673:
3349:
3747:
6526:
2996:
3358:
3344:{\displaystyle {\begin{aligned}\left\langle \varepsilon (t+dt)\right\rangle &=\left(1-\Gamma dt\right)\left\langle \varepsilon (t)\right\rangle +\Gamma dt\cdot \left\langle z\right\rangle \left(\left\langle \varepsilon _{i}\right\rangle +\left\langle \varepsilon _{j}\right\rangle \right)\\&=\left(1-\Gamma dt\right)\left\langle \varepsilon (t)\right\rangle +\Gamma dt\cdot {\dfrac {1}{2}}\left(\left\langle \varepsilon (t)\right\rangle +\left\langle \varepsilon (t)\right\rangle \right)\\&=\left\langle \varepsilon (t)\right\rangle \end{aligned}}}
3742:{\displaystyle {\begin{aligned}\left\langle \varepsilon ^{2}(t+dt)\right\rangle &=\left(1-\Gamma dt\right)\left\langle \varepsilon ^{2}(t)\right\rangle +\Gamma dt\cdot \left\langle z^{2}\right\rangle \left\langle \varepsilon _{i}^{2}+2\varepsilon _{i}\varepsilon _{j}+\varepsilon _{j}^{2}\right\rangle \\&=\left(1-\Gamma dt\right)\left\langle \varepsilon ^{2}(t)\right\rangle +\Gamma dt\cdot {\dfrac {1}{3}}\left(2\left\langle \varepsilon ^{2}(t)\right\rangle +2\left\langle \varepsilon (t)\right\rangle ^{2}\right)\end{aligned}}}
5999:
3993:
50:
31:
686:
974:
5697:
6009:
5382:
5118:
3756:
6581:
achieved by external driving. Ordering, or crystallization of granular materials has been observed to occur in periodically sheared as well as vibrated granular matter. In contrast to molecular systems, the positions of the individual particles can be tracked in the experiment. Computer simulations for a system of spherical grains reveal that homogeneous crystallization emerges at a volume fraction
5994:{\displaystyle \int _{0}^{\infty }{\dfrac {1}{T}}e^{-{\frac {\varepsilon }{T}}}\cdot e^{-\lambda \varepsilon }d\varepsilon ={\dfrac {1}{T}}\int _{0}^{\infty }e^{-\left(\lambda +{\frac {1}{T}}\right)\varepsilon }d\varepsilon =-{\dfrac {1}{T\left(\lambda +{\frac {1}{T}}\right)}}e^{-\left(\lambda +{\frac {1}{T}}\right)\varepsilon }|_{0}^{\infty }={\dfrac {1}{\lambda T+1}}=g(\lambda )}
4265:
6491:. The formation of ordered volumetric structures in granular materials is known as Granular Crystallisation, and involves a transition from a random packing of particles to an ordered packing such as hexagonal close-packed or body-centred cubic. This is most commonly observed in granular materials with narrow size distributions and uniform grain morphology.
5124:
4899:
4694:
4905:
3988:{\displaystyle {\dfrac {d\left\langle \varepsilon ^{2}\right\rangle }{dt}}=lim_{dt\rightarrow 0}{\dfrac {\left\langle \varepsilon ^{2}(t+dt)\right\rangle -\left\langle \varepsilon ^{2}(t)\right\rangle }{dt}}=-{\dfrac {\Gamma }{3}}\left\langle \varepsilon ^{2}\right\rangle +{\dfrac {2\Gamma }{3}}\left\langle \varepsilon \right\rangle ^{2}}
2915:
5620:
4102:
4531:
4111:
2300:
984:
regarded internal forces between granular particles as a friction process, and proposed the friction law, that the force of friction of solid particles is proportional to the normal pressure between them and the static friction coefficient is greater than the kinetic friction coefficient. He studied
4379:
2318:
nature of the collisions between grains. This clustering has some interesting consequences. For example, if a partially partitioned box of granular materials is vigorously shaken then grains will over time tend to collect in one of the partitions rather than spread evenly into both partitions as
6580:
When water or other liquids are cooled sufficiently slowly, randomly positioned molecules rearrange and solid crystals emerge and grow. A similar crystallisation process may occur in randomly packed granular materials. Unlike removing energy by cooling, crystallization in granular material is
6625:. Most of these methods consist of the statistical methods by which various statistical properties, derived from either point data or an image, are extracted and used to generate stochastic models of the granular medium. A recent and comprehensive review of such methods is available in
5377:{\displaystyle g\left(\lambda ,t+dt\right)=\left(1-\Gamma dt\right)g\left(\lambda ,t\right)+\Gamma dt\int _{0}^{1}{\underset {=g^{2}(\lambda z,t)}{\underbrace {\left\langle e^{-\lambda z\varepsilon _{i}(t)}\right\rangle \left\langle e^{-\lambda z\varepsilon _{j}(t)}\right\rangle } }}dz}
6482:
where Brazil nuts rise to the top of a packet of mixed nuts when shaken. The cause of this effect is that when shaken, granular (and some other) materials move in a circular pattern. some larger materials (Brazil nuts) get stuck while going down the circle and therefore stay on the
2678:
2309:
If the granular material is driven harder such that contacts between the grains become highly infrequent, the material enters a gaseous state. Correspondingly, one can define a granular temperature equal to the root mean square of grain velocity fluctuations that is analogous to
4703:
1510:
The given pressure equation does not account for boundary conditions, such as the ratio between the particle size to the radius of the silo. Since the internal stress of the material cannot be measured, Janssen's speculations have not been verified by any direct experiment.
4540:
5113:{\displaystyle \left\langle e^{-\lambda \varepsilon \left(t+dt\right)}\right\rangle =\left(1-\Gamma dt\right)\left\langle e^{-\lambda \varepsilon _{i}(t)}\right\rangle +\Gamma dt\left\langle e^{-\lambda z\left(\varepsilon _{i}(t)+\varepsilon _{j}(t)\right)}\right\rangle }
1115:
reaches a certain value, the force chains can break and the particles at the end of the chains on the surface begin to slide. Then, new force chains form until the shear stress is less than the critical value, and so the sandpile maintains a constant angle of repose.
1917:
5449:
6486:
The formation of structured surface or bulk patterns in vibrated granular layers. These patterns include but are not limited to stripes, squares and hexagons. These patterns are thought to be formed by fundamental excitations of the surface known as
1042:. When the sandpile slope reaches the maximum stable angle, the sand particles on the surface of the pile begin to fall. The process stops when the surface inclination angle is equal to the angle of repose. The difference between these two angles,
4002:
2166:
6816:
4422:
4260:{\displaystyle \left\langle \varepsilon ^{2}\right\rangle -2\left\langle \varepsilon \right\rangle ^{2}=\left(\left\langle \varepsilon ^{2}(0)\right\rangle -2\left\langle \varepsilon (0)\right\rangle ^{2}\right)e^{-{\frac {\Gamma }{3}}t}}
4276:
1124:
In 1895, H. A. Janssen discovered that in a vertical cylinder filled with particles, the pressure measured at the base of the cylinder does not depend on the height of the filling, unlike
Newtonian fluids at rest which follow
5686:
2666:
753:
collide). The constituents that compose granular material are large enough such that they are not subject to thermal motion fluctuations. Thus, the lower size limit for grains in granular material is about 1
6020:
and undergo a jamming transition which is thought of as a thermodynamic phase transition to a jammed state. The transition is from fluid-like phase to a solid-like phase and it is controlled by temperature,
4894:{\displaystyle e^{-\lambda \varepsilon _{i}(t+dt)}={\begin{cases}e^{-\lambda \varepsilon _{i}(t)}&1-\Gamma t\\e^{-\lambda z\left(\varepsilon _{i}(t)+\varepsilon _{j}(t)\right)}&\Gamma t\end{cases}}}
2597:
1627:
4689:{\displaystyle {\dfrac {d^{n}g}{d\lambda ^{n}}}=\left(-1\right)^{n}\int _{0}^{\infty }\varepsilon ^{n}e^{-\lambda \varepsilon }\rho (\varepsilon )d\varepsilon =\left\langle \varepsilon ^{n}\right\rangle }
906:
Granular materials also exhibit a wide range of pattern forming behaviors when excited (e.g. vibrated or allowed to flow). As such granular materials under excitation can be thought of as an example of a
6517:, the former being the most efficient for a higher density of the material and the motions of a lower intensity, and the latter for a lower density of the material and the motions of a higher intensity.
1806:
1332:
2910:{\displaystyle \varepsilon _{i}(t+dt)={\begin{cases}\varepsilon _{i}(t)&probability:\,1-\Gamma dt\\z\left(\varepsilon _{i}(t)+\varepsilon _{j}(t)\right)&probability:\,\Gamma dt\end{cases}}}
1276:
1089:
7386:
3363:
3001:
6333:
2502:
2462:
931:. In each particle are a lot of internal DOF. Consider inelastic collision between two particles - the energy from velocity as rigid body is transferred to microscopic internal DOF. We get “
5615:{\displaystyle \lambda g(\lambda )=\int _{0}^{\lambda }g^{2}(\delta )d\delta \Rightarrow \lambda g'(\lambda )+g(\lambda )=g^{2}(\lambda )\Rightarrow g(\lambda )={\dfrac {1}{\lambda T+1}}}
1459:
1750:
2119:
2040:
2159:
6611:
1568:
7173:
Jin, Weiwei; O’Hern, Corey S.; Radin, Charles; Shattuck, Mark D.; Swinney, Harry L. (18 December 2020). "Homogeneous
Crystallization in Cyclically Sheared Frictionless Grains".
6155:
4097:{\displaystyle {\dfrac {d\left\langle \varepsilon ^{2}\right\rangle }{dt}}=0\Rightarrow \left\langle \varepsilon ^{2}\right\rangle =2\left\langle \varepsilon \right\rangle ^{2}}
1970:, which describes the angle that if the tangential force falls within the friction cone the particles would still remain steady. It is determined by the coefficient of friction
5442:
2396:
2007:
1418:
1796:
6184:
2086:
1968:
6403:
6380:
6119:
4526:{\displaystyle {\dfrac {dg}{d\lambda }}=-\int _{0}^{\infty }\varepsilon e^{-\lambda \varepsilon }\rho (\varepsilon )d\varepsilon =-\left\langle \varepsilon \right\rangle }
1701:
1674:
1223:
1193:
1160:
5413:
1040:
1010:
2544:
6457:
6430:
6271:
6083:
2935:
2060:
1939:
4417:
1647:
1103:
which are networks of grains resting on one another. Between these chains are regions of low stress whose grains are shielded for the effects of the grains above by
6204:
6063:
2991:
2422:
1505:
6509:
Some of the pattern-forming behaviours have been possible to reproduce in computer simulations. There are two main computational approaches to such simulations,
6541:
during avalanching or when their surface is otherwise disturbed. Granular materials discharged from silos produce loud acoustic emissions in a process known as
6353:
6291:
6244:
6224:
6039:
2955:
2369:
2349:
2295:{\displaystyle {\frac {\dot {\Delta _{2}}}{\dot {\Delta _{1}}}}={\frac {{\dot {\varepsilon _{22}}}\ell _{2}}{{\dot {\varepsilon _{11}}}\ell _{1}}}=-\tan \beta }
1479:
7029:
Dai, Weijing; Reimann, Joerg; Hanaor, Dorian; Ferrero, Claudio; Gan, Yixiang (2019). "Modes of wall induced granular crystallisation in vibrational packing".
6186:
divides the state phase to jammed\unjammed region, which corresponds to granular solids\liquids respectively. For isotropically jammed granular system, when
4374:{\displaystyle g(\lambda )=\left\langle e^{-\lambda \varepsilon }\right\rangle =\int _{0}^{\infty }e^{-\lambda \varepsilon }\rho (\varepsilon )d\varepsilon }
7093:
6613:. The computer simulations identify the minimal ingredients necessary for granular crystallization. In particular, gravity and friction are not necessary.
923:
Granular matter is a system composed of many macroscopic particles. Microscopic particles (atoms\molecules) are described (in classical mechanics) by all
5627:
717:
6121:
plane and it is divided into a jammed state region and unjammed liquid state by a transition line. The phase diagram for granular matter lies in the
4269:
However, instead of characterizing the moments, we can analytically solve the energy distribution, from the moment generating function. Consider the
2602:
7235:
7097:
6626:
935:” - irreversible heat generation. The result is that without external driving, eventually all particles will stop moving. In macroscopic particles
1706:
Then stress on the boundary can be expressed as the concentrated force borne by individual particles. Under biaxial loading with uniform stress
6542:
6510:
6637:
algorithm by which the real shape of the particle can be captured and reproduced through the extracted statistics for particles' morphology.
903:
depending on the average energy per grain. However, in each of these states, granular materials also exhibit properties that are unique.
924:
868:
2549:
1573:
6474:
Excited granular matter is a rich pattern-forming system. Some of the pattern-forming behaviours seen in granular materials are:
1284:
710:
7380:
6935:
7077:
2993:(uniform distribution) and j is an index also randomly chosen from a uniform distribution. The average energy per particle:
1228:
1045:
821:
was originally stated for granular materials. Granular materials are commercially important in applications as diverse as
758:. On the upper size limit, the physics of granular materials may be applied to ice floes where the individual grains are
6296:
2121:
then the particles will begin sliding, resulting in changing the structure of the system and creating new force chains.
2467:
2427:
2323:, does not violate any thermodynamics principles since energy is constantly being lost from the system in the process.
1343:
7402:
6735:
6714:
703:
690:
1912:{\displaystyle {\frac {F_{11}}{F_{22}}}={\frac {\sigma _{11}\ell _{2}}{\sigma _{22}\ell _{1}}}=\tan(\theta +\beta )}
6463:
2672:
1527:
1570:, in vertical and horizontal directions respectively. The geometric characteristics of the system is described by
1423:
672:
1709:
17:
2091:
2012:
455:
7293:
6630:
6687:
2124:
1520:
6478:
The un-mixing or segregation of unlike grains under vibration and flow. An example of this is the so-called
7407:
630:
110:
6584:
1340:
The pressure in the granular material is then described in a different law, which accounts for saturation:
7328:
1533:
981:
814:
7334:
Lu, Kevin; et al. (November 2007). "Shear-weakening of the transitional regime for granular flow".
6124:
7336:
5418:
635:
260:
6525:
2374:
1973:
6432:
is constant for an infinite system, for a finite system boundary effects result in a distribution of
2311:
1755:
1526:
The mechanical properties of assembly of mono-dispersed particles in 2D can be analyzed based on the
525:
200:
6160:
4756:
2718:
2065:
1946:
745:
characterized by a loss of energy whenever the particles interact (the most common example would be
6817:"Origin of the long-ranged attraction or repulsion between intruders in a confined granular medium"
6514:
927:
of the system. Macroscopic particles are described only by DOF of the motion of each particle as a
520:
515:
41:
6989:
Rosato, A.; Strandburg, K.J.; Prinz, F.; Swendsen, R.H. (1987). "Why the Brazil Nuts are on Top".
6385:
6362:
6088:
1679:
1652:
1198:
1168:
1135:
985:
the collapse of piles of sand and found empirically two critical angles: the maximal stable angle
605:
6356:
5389:
1018:
988:
839:
are a special class of granular material due to their small particle size, which makes them more
210:
2507:
1099:: stress in a granular solid is not distributed uniformly but is conducted away along so-called
615:
6534:
6435:
6408:
6249:
5689:
1941:, the friction angle, is the angle between the contact force and the contact normal direction.
844:
600:
540:
510:
460:
180:
70:
6646:
6068:
2920:
2045:
1924:
640:
255:
240:
4387:
1649:, which describes the angle when the contact points begin the process of sliding. Denote by
1632:
7344:
7250:
7192:
7137:
6963:
6773:
6189:
6048:
2960:
1104:
936:
840:
230:
120:
6760:
Richard, P.; Nicodemi, Mario; Delannay, Renaud; Ribière, Philippe; Bideau, Daniel (2005).
2314:. Unlike conventional gases, granular materials will tend to cluster and clump due to the
813:. Research into granular materials is thus directly applicable and goes back at least to
8:
2401:
1484:
863:
470:
280:
130:
7348:
7254:
7196:
7141:
6967:
6777:
7360:
7274:
7216:
7182:
7127:
7114:
Rietz, Frank; Radin, Charles; Swinney, Harry L.; Schröter, Matthias (2 February 2018).
7056:
7038:
6891:
6844:
6797:
6709:(translated by A. Reisinger). November 1999, Springer-Verlag New York, Inc., New York,
6629:. Another alternative for building a pack of granular particles that recently has been
6563:
6338:
6276:
6229:
6209:
6024:
2940:
2504:. Now, randomly distribute the total energy between the two particles: choose randomly
2354:
2334:
2320:
1464:
836:
650:
610:
585:
333:
324:
6948:
6863:
1676:
the vertical direction, which is the direction of the major principal stress, and by
7376:
7266:
7220:
7208:
7155:
7060:
7006:
6931:
6883:
6848:
6836:
6789:
6731:
6710:
6661:
6656:
6479:
6017:
4270:
830:
739:
580:
315:
235:
7364:
7278:
6895:
6801:
6761:
6012:
Jamming during discharge of granular material is due to arch formation (red spheres)
7352:
7308:
7258:
7204:
7200:
7150:
7145:
7115:
7048:
6998:
6971:
6875:
6828:
6781:
6676:
6634:
888:
873:
285:
250:
245:
205:
175:
145:
105:
65:
7312:
6681:
7081:
7074:
6042:
5681:{\displaystyle \rho (\varepsilon )={\dfrac {1}{T}}e^{-{\frac {\varepsilon }{T}}}}
1013:
595:
545:
415:
170:
82:
7002:
6707:
Sands, Powders, and Grains: An
Introduction to the Physics of Granular Materials
1703:
the horizontal direction, which is the direction of the minor principal stress.
6879:
6651:
908:
822:
810:
677:
645:
625:
620:
575:
495:
430:
328:
215:
60:
7356:
7075:
Computer
Simulation of Granular Matter: A Study of An Industrial Grinding Mill
7052:
2661:{\displaystyle \left(1-z\right)\left(\varepsilon _{i}+\varepsilon _{j}\right)}
1523:
effect on shear strength in shear tests and proposed a relation between them.
7396:
7270:
6840:
6815:
Dhiman, Manish; Kumar, Sonu; Reddy, K. Anki; Gupta, Raghvendra (March 2020).
6538:
6537:, exhibit squeaking when walked upon. Some desert dunes are known to exhibit
1129:'s law. Janssen suggested a simplified model with the following assumptions:
912:
763:
356:
337:
319:
220:
140:
550:
7212:
7159:
7010:
6887:
6793:
6671:
6666:
6495:
6466:
of jamming allows one to produce simulated jammed granular configurations.
1126:
1112:
880:
and are the second-most manipulated material in industry (the first one is
767:
570:
560:
530:
490:
485:
465:
310:
290:
150:
30:
6832:
7262:
6554:
5692:) by taking its Laplace transform and calculate the generating function:
2315:
1100:
1096:
932:
826:
802:
782:
750:
655:
590:
565:
535:
480:
475:
407:
49:
2319:
would happen in a conventional gas. This effect, known as the granular
1092:
928:
755:
500:
342:
135:
6975:
6947:
Kansal, Anuraag R.; Torquato, Salvatore; Stillinger, Frank H. (2002).
866:
was an early pioneer of the physics of granular matter and whose book
6785:
6503:
2398:. At some constant rate per unit time, randomly choose two particles
973:
859:
555:
505:
378:
225:
125:
7372:
7234:
Tahmasebi, Pejman; Sahimi, Muhammad; Andrade, José E. (2017-01-01).
2161:, the horizontal and vertical displacements respectively satisfies:
911:. They also display fluid-based instabilities and phenomena such as
7187:
7132:
7043:
6566:
6488:
6008:
818:
771:
746:
742:
115:
6759:
6569:
are made to adhere to form larger, multiparticle entities called
855:
759:
435:
420:
383:
374:
369:
6616:
2592:{\displaystyle z\left(\varepsilon _{i}+\varepsilon _{j}\right)}
1622:{\displaystyle \alpha =\arctan({\frac {\ell _{1}}{\ell _{2}}})}
942:
When a matter is dilute and dynamic (driven) then it is called
896:
877:
798:
388:
364:
95:
6988:
892:
887:
In some sense, granular materials do not constitute a single
881:
735:
393:
90:
2546:
so that the first particle, after the collision, has energy
1337:
4) The density of the material is constant over all depths.
6949:"Computer Generation of Dense Polydisperse Sphere Packings"
6499:
4887:
2903:
1108:
977:
Chain of transmission of stress forces in a granular medium
806:
794:
790:
786:
778:
6862:
Kumar, Sonu; Dhiman, Manish; Reddy, K. Anki (2019-01-14).
6382:
and there are some divergent characteristics lengths when
1481:
is the radius of the cylinder, and at the top of the silo
872:
remains an important reference to this day. According to
7113:
6085:. The normal phase diagram of glass transition is in the
4106:
Solving the differential equation for the second moment:
1334:
sustains the vertical load at the contact with the wall;
1327:{\displaystyle \mu ={\frac {\sigma _{rz}}{\sigma _{rr}}}}
900:
848:
100:
7373:
The new physical-mechanical theory of granular materials
6946:
7028:
876:
Patrick
Richard, "Granular materials are ubiquitous in
27:
Conglomeration of discrete solid, macroscopic particles
7233:
7172:
1271:{\displaystyle K={\frac {\sigma _{rr}}{\sigma _{zz}}}}
1084:{\displaystyle \Delta \theta =\theta _{m}-\theta _{r}}
7387:
Modelling and
Numerics of Kinetic Dissipative Systems
7098:
Intruder dynamics on vibrofluidized granular surfaces
6814:
6587:
6438:
6411:
6388:
6365:
6341:
6299:
6279:
6252:
6232:
6212:
6192:
6163:
6127:
6091:
6071:
6051:
6027:
5954:
5855:
5777:
5717:
5700:
5647:
5630:
5590:
5452:
5421:
5392:
5127:
4908:
4706:
4545:
4543:
4427:
4425:
4390:
4279:
4114:
4007:
4005:
3951:
3918:
3826:
3761:
3759:
3647:
3361:
3237:
2999:
2963:
2943:
2923:
2681:
2605:
2552:
2510:
2470:
2430:
2404:
2377:
2357:
2337:
2169:
2127:
2094:
2068:
2048:
2015:
1976:
1949:
1927:
1809:
1758:
1712:
1682:
1655:
1635:
1576:
1536:
1487:
1467:
1426:
1346:
1287:
1231:
1201:
1171:
1138:
1095:
of granular materials. This phenomenon is due to the
1048:
1021:
991:
949:
When a matter is dense and static, then it is called
6762:"Slow relaxation and compaction of granular systems"
956:
When the density is intermediate, then it is called
6533:Some beach sands, such as those of the aptly named
1514:
6605:
6451:
6424:
6397:
6374:
6359:: the bulk modulus shows a power law scaling with
6347:
6328:{\displaystyle \Delta \phi \equiv \phi -\phi _{c}}
6327:
6285:
6265:
6246:point corresponds to the critical volume fraction
6238:
6218:
6198:
6178:
6149:
6113:
6077:
6057:
6033:
5993:
5680:
5614:
5436:
5407:
5376:
5112:
4893:
4688:
4525:
4411:
4373:
4259:
4096:
3987:
3741:
3343:
2985:
2949:
2929:
2909:
2660:
2591:
2538:
2496:
2456:
2416:
2390:
2363:
2343:
2294:
2153:
2113:
2080:
2054:
2034:
2001:
1962:
1933:
1911:
1790:
1744:
1695:
1668:
1641:
1621:
1562:
1499:
1473:
1453:
1412:
1326:
1270:
1217:
1187:
1154:
1091:, is the Bagnold angle, which is a measure of the
1083:
1034:
1004:
2497:{\displaystyle \varepsilon _{i}+\varepsilon _{j}}
2457:{\displaystyle \varepsilon _{i},\varepsilon _{j}}
7394:
7102:Materials Research Society Symposium Proceedings
6861:
7329:Fundamentals of Particle Technology – free book
7236:"Image-Based Modeling of Granular Porous Media"
6684:: one model of the rheology of a granular flow.
7084:, Thesis, Univ. Western Ontario, Canada, 2004.
3751:Now the time derivative of the second moment:
7294:"Packing of discrete and irregular particles"
711:
6617:Computational modeling of granular materials
6335:. The behavior of granular systems near the
6226:, the bulk and shear moduli approach 0. The
2042:. Once stress is applied to the system then
1454:{\displaystyle \lambda ={\frac {R}{2\mu K}}}
1195:, is proportional to the vertical pressure
7024:
7022:
7020:
1745:{\displaystyle \sigma _{12}=\sigma _{21}=0}
7389:, Nova Science Publishers, New York, 2006.
6748:The physics of blown sand and desert dunes
2114:{\displaystyle \theta \geq \theta _{\mu }}
2035:{\displaystyle \theta \leq \theta _{\mu }}
869:The Physics of Blown Sand and Desert Dunes
718:
704:
48:
7291:
7186:
7149:
7131:
7042:
6926:Haye Hinrichsen, Dietrich E. Wolf (eds),
2890:
2778:
7017:
6930:. 2004, Wiley-VCH Verlag GmbH & Co.
6524:
6355:point was empirically found to resemble
6007:
1281:3) The wall friction static coefficient
972:
891:but have characteristics reminiscent of
777:Some examples of granular materials are
29:
7116:"Nucleation in Sheared Granular Matter"
6911:
6562:is the act or process in which primary
2154:{\displaystyle \Delta _{1},\Delta _{2}}
1162:, is constant in the horizontal plane;
968:
14:
7395:
7385:Pareschi, L., Russo, G., Toscani, G.,
6016:Granular systems are known to exhibit
946:and dissipation phenomenon dominates.
6157:plane, and the critical stress curve
6003:
6907:
6905:
6606:{\displaystyle \phi =0.646\pm 0.001}
6469:
6520:
6206:is reduced around a certain point,
1563:{\displaystyle \ell _{1},\ell _{2}}
963:
24:
7333:
6621:Several methods are available for
6575:
6389:
6366:
6300:
6164:
6150:{\displaystyle \phi ^{-1}-\Sigma }
6144:
6072:
5945:
5798:
5711:
5212:
5173:
5026:
4969:
4878:
4799:
4616:
4464:
4332:
4244:
3957:
3920:
3634:
3588:
3468:
3422:
3224:
3185:
3092:
3053:
2924:
2891:
2785:
2192:
2175:
2142:
2129:
1367:
1049:
953:and jamming phenomenon dominates.
25:
7419:
7322:
7292:Tahmasebi, Pejman (August 2018).
6902:
6864:"Magnus effect in granular media"
5437:{\displaystyle \delta =\lambda z}
2304:
1519:In the early 1960s, Rowe studied
1119:
6464:Lubachevsky-Stillinger algorithm
6293:, the critical volume fraction,
2391:{\displaystyle \varepsilon _{i}}
2002:{\displaystyle \mu =tg\phi _{u}}
1528:representative elementary volume
1515:Rowe Stress - Dilatancy Relation
1413:{\displaystyle p(z)=p_{\infty }}
734:is a conglomeration of discrete
685:
684:
671:
7285:
7227:
7166:
7107:
7087:
7067:
6982:
6956:The Journal of Chemical Physics
6728:Principles of powder technology
6273:. Define the distance to point
1791:{\displaystyle F_{12}=F_{21}=0}
7205:10.1103/PhysRevLett.125.258003
7151:10.1103/PhysRevLett.120.055701
6940:
6920:
6914:"Mechanics of Granular Matter"
6855:
6808:
6753:
6740:
6730:, John Wiley & Sons, 1997
6720:
6699:
6623:modeling of granular materials
6548:
6179:{\displaystyle \Sigma (\phi )}
6173:
6167:
5988:
5982:
5935:
5640:
5634:
5583:
5577:
5571:
5568:
5562:
5546:
5540:
5531:
5525:
5511:
5502:
5496:
5465:
5459:
5402:
5396:
5362:
5347:
5318:
5312:
5275:
5269:
5096:
5090:
5074:
5068:
5014:
5008:
4866:
4860:
4844:
4838:
4786:
4780:
4743:
4728:
4656:
4650:
4497:
4491:
4400:
4394:
4362:
4356:
4289:
4283:
4215:
4209:
4186:
4180:
4049:
3891:
3885:
3859:
3844:
3817:
3716:
3710:
3687:
3681:
3623:
3617:
3457:
3451:
3396:
3381:
3329:
3323:
3292:
3286:
3267:
3261:
3213:
3207:
3081:
3075:
3027:
3012:
2844:
2838:
2822:
2816:
2737:
2731:
2707:
2692:
2081:{\displaystyle \alpha ,\beta }
1963:{\displaystyle \theta _{\mu }}
1906:
1894:
1616:
1589:
1407:
1404:
1387:
1372:
1356:
1350:
918:
34:Examples of granular materials
13:
1:
7313:10.1016/j.compgeo.2018.03.011
6928:The Physics of Granular Media
6916:. Southampton, UK: WIT Press.
6693:
6688:Dilatancy (granular material)
2326:
1165:2) The horizontal pressure,
770:with individual grains being
7243:Geophysical Research Letters
6398:{\displaystyle \Delta \phi }
6375:{\displaystyle \Delta \phi }
6114:{\displaystyle \phi ^{-1}-T}
1696:{\displaystyle \sigma _{22}}
1669:{\displaystyle \sigma _{11}}
1218:{\displaystyle \sigma _{zz}}
1188:{\displaystyle \sigma _{rr}}
1155:{\displaystyle \sigma _{zz}}
7:
7003:10.1103/physrevlett.58.1038
6640:
5408:{\displaystyle g(\lambda )}
1035:{\displaystyle \theta _{r}}
1005:{\displaystyle \theta _{m}}
815:Charles-Augustin de Coulomb
10:
7424:
6880:10.1103/PhysRevE.99.012902
6821:Journal of Fluid Mechanics
6627:Tahmasebi and other (2017)
6552:
2539:{\displaystyle z\in \left}
2062:gradually increases while
1132:1) The vertical pressure,
261:Spin gapless semiconductor
7357:10.1017/S0022112007007331
7301:Computers and Geotechnics
7053:10.1007/s10035-019-0876-8
6452:{\displaystyle \phi _{c}}
6425:{\displaystyle \phi _{c}}
6266:{\displaystyle \phi _{c}}
5415:with change of variables
2312:thermodynamic temperature
201:Electronic band structure
7403:Granularity of materials
2957:is randomly picked from
2088:remains unchanged. When
1530:, with typical lengths,
111:Bose–Einstein condensate
42:Condensed matter physics
7175:Physical Review Letters
7120:Physical Review Letters
6991:Physical Review Letters
6405:approaches zero. While
6357:second-order transition
6078:{\displaystyle \Sigma }
2937:is the collision rate,
2930:{\displaystyle \Gamma }
2055:{\displaystyle \theta }
1934:{\displaystyle \theta }
7104:1152E, TT03-01 (2009).
6607:
6530:
6494:The formation of sand
6453:
6426:
6399:
6376:
6349:
6329:
6287:
6267:
6240:
6220:
6200:
6180:
6151:
6115:
6079:
6059:
6035:
6013:
5995:
5690:Boltzmann Distribution
5682:
5616:
5438:
5409:
5378:
5114:
4895:
4690:
4527:
4413:
4412:{\displaystyle g(0)=1}
4375:
4261:
4098:
3989:
3743:
3345:
2987:
2951:
2931:
2911:
2662:
2593:
2540:
2498:
2458:
2418:
2392:
2365:
2345:
2296:
2155:
2115:
2082:
2056:
2036:
2003:
1964:
1935:
1913:
1801:At equilibrium state:
1792:
1746:
1697:
1670:
1643:
1642:{\displaystyle \beta }
1623:
1564:
1501:
1475:
1455:
1414:
1328:
1278:is constant in space;
1272:
1219:
1189:
1156:
1085:
1036:
1006:
978:
35:
6912:Qicheng, Sun (2013).
6833:10.1017/jfm.2019.1035
6647:Aggregate (composite)
6608:
6528:
6454:
6427:
6400:
6377:
6350:
6330:
6288:
6268:
6241:
6221:
6201:
6199:{\displaystyle \phi }
6181:
6152:
6116:
6080:
6060:
6058:{\displaystyle \phi }
6036:
6011:
5996:
5683:
5617:
5439:
5410:
5379:
5115:
4896:
4691:
4528:
4414:
4376:
4262:
4099:
3990:
3744:
3346:
2988:
2986:{\displaystyle \left}
2952:
2932:
2912:
2663:
2594:
2541:
2499:
2459:
2419:
2393:
2366:
2346:
2297:
2156:
2116:
2083:
2057:
2037:
2004:
1965:
1936:
1914:
1793:
1747:
1698:
1671:
1644:
1624:
1565:
1502:
1476:
1456:
1415:
1329:
1273:
1220:
1190:
1157:
1086:
1037:
1007:
976:
256:Topological insulator
33:
7263:10.1002/2017GL073938
7249:(10): 2017GL073938.
6746:Bagnold, R.A. 1941.
6726:Rodhes, M (editor),
6585:
6436:
6409:
6386:
6363:
6339:
6297:
6277:
6250:
6230:
6210:
6190:
6161:
6125:
6089:
6069:
6065:, and shear stress,
6049:
6025:
5698:
5628:
5450:
5419:
5390:
5125:
4906:
4704:
4541:
4423:
4388:
4277:
4112:
4003:
3757:
3359:
2997:
2961:
2941:
2921:
2679:
2673:stochastic evolution
2603:
2550:
2508:
2468:
2464:and compute the sum
2428:
2402:
2375:
2355:
2351:particles, particle
2335:
2167:
2125:
2092:
2066:
2046:
2013:
1974:
1947:
1925:
1807:
1756:
1710:
1680:
1653:
1633:
1574:
1534:
1485:
1465:
1424:
1344:
1285:
1229:
1199:
1169:
1136:
1046:
1019:
989:
969:Coulomb friction Law
937:thermal fluctuations
274:Electronic phenomena
121:Fermionic condensate
7408:Discrete-phase flow
7349:2007JFM...587..347L
7255:2017GeoRL..44.4738T
7197:2020PhRvL.125y8003J
7142:2018PhRvL.120e5701R
7094:A. D. Wissner-Gross
6968:2002JChPh.117.8212K
6778:2005NatMa...4..121R
5949:
5802:
5715:
5485:
5235:
4620:
4468:
4336:
3561:
3517:
3353:The second moment:
2417:{\displaystyle i,j}
1500:{\displaystyle z=0}
864:Ralph Alger Bagnold
281:Quantum Hall effect
7375:. 2009, Homonnai,
7080:2011-08-18 at the
6750:. London: Methuen,
6603:
6531:
6449:
6422:
6395:
6372:
6345:
6325:
6283:
6263:
6236:
6216:
6196:
6176:
6147:
6111:
6075:
6055:
6031:
6014:
6004:Jamming transition
5991:
5974:
5933:
5892:
5788:
5786:
5726:
5701:
5678:
5656:
5624:We will show that
5612:
5610:
5471:
5434:
5405:
5374:
5366:
5331:
5221:
5110:
4891:
4886:
4686:
4606:
4578:
4535:the n derivative:
4523:
4454:
4446:
4409:
4371:
4322:
4257:
4094:
4041:
3985:
3965:
3927:
3909:
3795:
3739:
3737:
3656:
3547:
3503:
3341:
3339:
3246:
2983:
2947:
2927:
2907:
2902:
2658:
2589:
2536:
2494:
2454:
2414:
2388:
2361:
2341:
2292:
2151:
2111:
2078:
2052:
2032:
1999:
1960:
1931:
1909:
1788:
1742:
1693:
1666:
1639:
1619:
1560:
1497:
1471:
1451:
1410:
1324:
1268:
1215:
1185:
1152:
1081:
1032:
1002:
979:
874:material scientist
678:Physics portal
36:
7381:978-963-8343-87-1
6976:10.1063/1.1511510
6936:978-3-527-60362-6
6868:Physical Review E
6662:Soil liquefaction
6657:Random close pack
6480:Brazil nut effect
6470:Pattern formation
6459:over some range.
6348:{\displaystyle J}
6286:{\displaystyle J}
6239:{\displaystyle J}
6219:{\displaystyle J}
6034:{\displaystyle T}
5973:
5921:
5891:
5883:
5830:
5785:
5744:
5725:
5674:
5655:
5609:
5239:
5237:
4577:
4445:
4271:Laplace transform
4250:
4040:
3997:In steady state:
3964:
3926:
3908:
3794:
3655:
3245:
2950:{\displaystyle z}
2599:, and the second
2364:{\displaystyle i}
2344:{\displaystyle N}
2275:
2261:
2230:
2206:
2204:
2187:
1883:
1832:
1629:and the variable
1614:
1474:{\displaystyle R}
1449:
1322:
1266:
831:energy production
732:granular material
728:
727:
426:Granular material
194:Electronic phases
16:(Redirected from
7415:
7368:
7317:
7316:
7298:
7289:
7283:
7282:
7240:
7231:
7225:
7224:
7190:
7170:
7164:
7163:
7153:
7135:
7111:
7105:
7091:
7085:
7071:
7065:
7064:
7046:
7026:
7015:
7014:
6986:
6980:
6979:
6953:
6944:
6938:
6924:
6918:
6917:
6909:
6900:
6899:
6859:
6853:
6852:
6812:
6806:
6805:
6786:10.1038/nmat1300
6766:Nature Materials
6757:
6751:
6744:
6738:
6724:
6718:
6703:
6677:Paste (rheology)
6633:is based on the
6612:
6610:
6609:
6604:
6521:Acoustic effects
6458:
6456:
6455:
6450:
6448:
6447:
6431:
6429:
6428:
6423:
6421:
6420:
6404:
6402:
6401:
6396:
6381:
6379:
6378:
6373:
6354:
6352:
6351:
6346:
6334:
6332:
6331:
6326:
6324:
6323:
6292:
6290:
6289:
6284:
6272:
6270:
6269:
6264:
6262:
6261:
6245:
6243:
6242:
6237:
6225:
6223:
6222:
6217:
6205:
6203:
6202:
6197:
6185:
6183:
6182:
6177:
6156:
6154:
6153:
6148:
6140:
6139:
6120:
6118:
6117:
6112:
6104:
6103:
6084:
6082:
6081:
6076:
6064:
6062:
6061:
6056:
6040:
6038:
6037:
6032:
6000:
5998:
5997:
5992:
5975:
5972:
5955:
5948:
5943:
5938:
5932:
5931:
5927:
5923:
5922:
5914:
5893:
5890:
5889:
5885:
5884:
5876:
5856:
5841:
5840:
5836:
5832:
5831:
5823:
5801:
5796:
5787:
5778:
5766:
5765:
5747:
5746:
5745:
5737:
5727:
5718:
5714:
5709:
5687:
5685:
5684:
5679:
5677:
5676:
5675:
5667:
5657:
5648:
5621:
5619:
5618:
5613:
5611:
5608:
5591:
5561:
5560:
5524:
5495:
5494:
5484:
5479:
5443:
5441:
5440:
5435:
5414:
5412:
5411:
5406:
5383:
5381:
5380:
5375:
5367:
5365:
5346:
5345:
5332:
5327:
5326:
5322:
5321:
5311:
5310:
5283:
5279:
5278:
5268:
5267:
5234:
5229:
5208:
5204:
5186:
5182:
5158:
5154:
5119:
5117:
5116:
5111:
5109:
5105:
5104:
5103:
5099:
5089:
5088:
5067:
5066:
5022:
5018:
5017:
5007:
5006:
4982:
4978:
4954:
4950:
4949:
4948:
4944:
4900:
4898:
4897:
4892:
4890:
4889:
4875:
4874:
4873:
4869:
4859:
4858:
4837:
4836:
4790:
4789:
4779:
4778:
4747:
4746:
4727:
4726:
4695:
4693:
4692:
4687:
4685:
4681:
4680:
4646:
4645:
4630:
4629:
4619:
4614:
4605:
4604:
4599:
4595:
4579:
4576:
4575:
4574:
4561:
4557:
4556:
4546:
4532:
4530:
4529:
4524:
4522:
4487:
4486:
4467:
4462:
4447:
4444:
4436:
4428:
4418:
4416:
4415:
4410:
4380:
4378:
4377:
4372:
4352:
4351:
4335:
4330:
4318:
4314:
4313:
4266:
4264:
4263:
4258:
4256:
4255:
4251:
4243:
4233:
4229:
4228:
4227:
4222:
4218:
4193:
4189:
4179:
4178:
4156:
4155:
4150:
4132:
4128:
4127:
4103:
4101:
4100:
4095:
4093:
4092:
4087:
4069:
4065:
4064:
4042:
4039:
4031:
4030:
4026:
4025:
4008:
3994:
3992:
3991:
3986:
3984:
3983:
3978:
3966:
3960:
3952:
3946:
3942:
3941:
3928:
3919:
3910:
3907:
3899:
3898:
3894:
3884:
3883:
3866:
3862:
3843:
3842:
3827:
3824:
3823:
3796:
3793:
3785:
3784:
3780:
3779:
3762:
3748:
3746:
3745:
3740:
3738:
3734:
3730:
3729:
3728:
3723:
3719:
3694:
3690:
3680:
3679:
3657:
3648:
3630:
3626:
3616:
3615:
3601:
3597:
3570:
3566:
3562:
3560:
3555:
3543:
3542:
3533:
3532:
3516:
3511:
3497:
3493:
3492:
3464:
3460:
3450:
3449:
3435:
3431:
3403:
3399:
3380:
3379:
3350:
3348:
3347:
3342:
3340:
3336:
3332:
3308:
3304:
3300:
3299:
3295:
3274:
3270:
3247:
3238:
3220:
3216:
3198:
3194:
3167:
3163:
3159:
3158:
3154:
3153:
3137:
3133:
3132:
3114:
3088:
3084:
3066:
3062:
3034:
3030:
2992:
2990:
2989:
2984:
2982:
2978:
2956:
2954:
2953:
2948:
2936:
2934:
2933:
2928:
2916:
2914:
2913:
2908:
2906:
2905:
2851:
2847:
2837:
2836:
2815:
2814:
2730:
2729:
2691:
2690:
2667:
2665:
2664:
2659:
2657:
2653:
2652:
2651:
2639:
2638:
2624:
2620:
2598:
2596:
2595:
2590:
2588:
2584:
2583:
2582:
2570:
2569:
2545:
2543:
2542:
2537:
2535:
2531:
2503:
2501:
2500:
2495:
2493:
2492:
2480:
2479:
2463:
2461:
2460:
2455:
2453:
2452:
2440:
2439:
2423:
2421:
2420:
2415:
2397:
2395:
2394:
2389:
2387:
2386:
2370:
2368:
2367:
2362:
2350:
2348:
2347:
2342:
2301:
2299:
2298:
2293:
2276:
2274:
2273:
2272:
2263:
2262:
2257:
2256:
2247:
2243:
2242:
2241:
2232:
2231:
2226:
2225:
2216:
2212:
2207:
2205:
2200:
2199:
2190:
2188:
2183:
2182:
2173:
2171:
2160:
2158:
2157:
2152:
2150:
2149:
2137:
2136:
2120:
2118:
2117:
2112:
2110:
2109:
2087:
2085:
2084:
2079:
2061:
2059:
2058:
2053:
2041:
2039:
2038:
2033:
2031:
2030:
2008:
2006:
2005:
2000:
1998:
1997:
1969:
1967:
1966:
1961:
1959:
1958:
1940:
1938:
1937:
1932:
1918:
1916:
1915:
1910:
1884:
1882:
1881:
1880:
1871:
1870:
1860:
1859:
1858:
1849:
1848:
1838:
1833:
1831:
1830:
1821:
1820:
1811:
1797:
1795:
1794:
1789:
1781:
1780:
1768:
1767:
1751:
1749:
1748:
1743:
1735:
1734:
1722:
1721:
1702:
1700:
1699:
1694:
1692:
1691:
1675:
1673:
1672:
1667:
1665:
1664:
1648:
1646:
1645:
1640:
1628:
1626:
1625:
1620:
1615:
1613:
1612:
1603:
1602:
1593:
1569:
1567:
1566:
1561:
1559:
1558:
1546:
1545:
1506:
1504:
1503:
1498:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1450:
1448:
1434:
1419:
1417:
1416:
1411:
1400:
1371:
1370:
1333:
1331:
1330:
1325:
1323:
1321:
1320:
1308:
1307:
1295:
1277:
1275:
1274:
1269:
1267:
1265:
1264:
1252:
1251:
1239:
1224:
1222:
1221:
1216:
1214:
1213:
1194:
1192:
1191:
1186:
1184:
1183:
1161:
1159:
1158:
1153:
1151:
1150:
1090:
1088:
1087:
1082:
1080:
1079:
1067:
1066:
1041:
1039:
1038:
1033:
1031:
1030:
1012:and the minimum
1011:
1009:
1008:
1003:
1001:
1000:
964:Static behaviors
939:are irrelevant.
843:and more easily
720:
713:
706:
693:
688:
687:
680:
676:
675:
286:Spin Hall effect
176:Phase transition
146:Luttinger liquid
83:States of matter
66:Phase transition
52:
38:
37:
21:
7423:
7422:
7418:
7417:
7416:
7414:
7413:
7412:
7393:
7392:
7325:
7320:
7296:
7290:
7286:
7238:
7232:
7228:
7171:
7167:
7112:
7108:
7092:
7088:
7082:Wayback Machine
7073:John J. Drozd,
7072:
7068:
7031:Granular Matter
7027:
7018:
6997:(10): 1038–41.
6987:
6983:
6951:
6945:
6941:
6925:
6921:
6910:
6903:
6860:
6856:
6813:
6809:
6758:
6754:
6745:
6741:
6725:
6721:
6704:
6700:
6696:
6643:
6619:
6586:
6583:
6582:
6578:
6576:Crystallization
6557:
6551:
6523:
6472:
6443:
6439:
6437:
6434:
6433:
6416:
6412:
6410:
6407:
6406:
6387:
6384:
6383:
6364:
6361:
6360:
6340:
6337:
6336:
6319:
6315:
6298:
6295:
6294:
6278:
6275:
6274:
6257:
6253:
6251:
6248:
6247:
6231:
6228:
6227:
6211:
6208:
6207:
6191:
6188:
6187:
6162:
6159:
6158:
6132:
6128:
6126:
6123:
6122:
6096:
6092:
6090:
6087:
6086:
6070:
6067:
6066:
6050:
6047:
6046:
6043:volume fraction
6026:
6023:
6022:
6006:
5959:
5953:
5944:
5939:
5934:
5913:
5906:
5902:
5898:
5894:
5875:
5868:
5864:
5860:
5854:
5822:
5815:
5811:
5807:
5803:
5797:
5792:
5776:
5755:
5751:
5736:
5732:
5728:
5716:
5710:
5705:
5699:
5696:
5695:
5666:
5662:
5658:
5646:
5629:
5626:
5625:
5595:
5589:
5556:
5552:
5517:
5490:
5486:
5480:
5475:
5451:
5448:
5447:
5420:
5417:
5416:
5391:
5388:
5387:
5341:
5337:
5333:
5306:
5302:
5292:
5288:
5284:
5263:
5259:
5249:
5245:
5241:
5240:
5238:
5236:
5230:
5225:
5194:
5190:
5166:
5162:
5135:
5131:
5126:
5123:
5122:
5084:
5080:
5062:
5058:
5057:
5053:
5043:
5039:
5035:
5002:
4998:
4991:
4987:
4983:
4962:
4958:
4931:
4927:
4917:
4913:
4909:
4907:
4904:
4903:
4885:
4884:
4876:
4854:
4850:
4832:
4828:
4827:
4823:
4813:
4809:
4806:
4805:
4791:
4774:
4770:
4763:
4759:
4752:
4751:
4722:
4718:
4711:
4707:
4705:
4702:
4701:
4676:
4672:
4668:
4635:
4631:
4625:
4621:
4615:
4610:
4600:
4588:
4584:
4583:
4570:
4566:
4562:
4552:
4548:
4547:
4544:
4542:
4539:
4538:
4512:
4476:
4472:
4463:
4458:
4437:
4429:
4426:
4424:
4421:
4420:
4389:
4386:
4385:
4341:
4337:
4331:
4326:
4303:
4299:
4295:
4278:
4275:
4274:
4242:
4238:
4234:
4223:
4205:
4201:
4200:
4174:
4170:
4169:
4165:
4164:
4160:
4151:
4140:
4139:
4123:
4119:
4115:
4113:
4110:
4109:
4088:
4077:
4076:
4060:
4056:
4052:
4032:
4021:
4017:
4013:
4009:
4006:
4004:
4001:
4000:
3979:
3968:
3967:
3953:
3950:
3937:
3933:
3929:
3917:
3900:
3879:
3875:
3874:
3870:
3838:
3834:
3833:
3829:
3828:
3825:
3810:
3806:
3786:
3775:
3771:
3767:
3763:
3760:
3758:
3755:
3754:
3736:
3735:
3724:
3706:
3702:
3701:
3675:
3671:
3670:
3666:
3662:
3658:
3646:
3611:
3607:
3606:
3602:
3581:
3577:
3568:
3567:
3556:
3551:
3538:
3534:
3528:
3524:
3512:
3507:
3502:
3498:
3488:
3484:
3480:
3445:
3441:
3440:
3436:
3415:
3411:
3404:
3375:
3371:
3370:
3366:
3362:
3360:
3357:
3356:
3338:
3337:
3319:
3315:
3306:
3305:
3282:
3278:
3257:
3253:
3252:
3248:
3236:
3203:
3199:
3178:
3174:
3165:
3164:
3149:
3145:
3141:
3128:
3124:
3120:
3119:
3115:
3104:
3071:
3067:
3046:
3042:
3035:
3008:
3004:
3000:
2998:
2995:
2994:
2968:
2964:
2962:
2959:
2958:
2942:
2939:
2938:
2922:
2919:
2918:
2901:
2900:
2852:
2832:
2828:
2810:
2806:
2805:
2801:
2795:
2794:
2740:
2725:
2721:
2714:
2713:
2686:
2682:
2680:
2677:
2676:
2647:
2643:
2634:
2630:
2629:
2625:
2610:
2606:
2604:
2601:
2600:
2578:
2574:
2565:
2561:
2560:
2556:
2551:
2548:
2547:
2521:
2517:
2509:
2506:
2505:
2488:
2484:
2475:
2471:
2469:
2466:
2465:
2448:
2444:
2435:
2431:
2429:
2426:
2425:
2403:
2400:
2399:
2382:
2378:
2376:
2373:
2372:
2356:
2353:
2352:
2336:
2333:
2332:
2329:
2321:Maxwell's demon
2307:
2268:
2264:
2252:
2248:
2246:
2245:
2244:
2237:
2233:
2221:
2217:
2215:
2214:
2213:
2211:
2195:
2191:
2189:
2178:
2174:
2172:
2170:
2168:
2165:
2164:
2145:
2141:
2132:
2128:
2126:
2123:
2122:
2105:
2101:
2093:
2090:
2089:
2067:
2064:
2063:
2047:
2044:
2043:
2026:
2022:
2014:
2011:
2010:
1993:
1989:
1975:
1972:
1971:
1954:
1950:
1948:
1945:
1944:
1926:
1923:
1922:
1876:
1872:
1866:
1862:
1861:
1854:
1850:
1844:
1840:
1839:
1837:
1826:
1822:
1816:
1812:
1810:
1808:
1805:
1804:
1776:
1772:
1763:
1759:
1757:
1754:
1753:
1730:
1726:
1717:
1713:
1711:
1708:
1707:
1687:
1683:
1681:
1678:
1677:
1660:
1656:
1654:
1651:
1650:
1634:
1631:
1630:
1608:
1604:
1598:
1594:
1592:
1575:
1572:
1571:
1554:
1550:
1541:
1537:
1535:
1532:
1531:
1517:
1486:
1483:
1482:
1466:
1463:
1462:
1438:
1433:
1425:
1422:
1421:
1396:
1366:
1362:
1345:
1342:
1341:
1313:
1309:
1300:
1296:
1294:
1286:
1283:
1282:
1257:
1253:
1244:
1240:
1238:
1230:
1227:
1226:
1206:
1202:
1200:
1197:
1196:
1176:
1172:
1170:
1167:
1166:
1143:
1139:
1137:
1134:
1133:
1122:
1075:
1071:
1062:
1058:
1047:
1044:
1043:
1026:
1022:
1020:
1017:
1016:
1014:angle of repose
996:
992:
990:
987:
986:
971:
966:
958:granular liquid
921:
889:phase of matter
819:law of friction
724:
683:
670:
669:
662:
661:
660:
450:
442:
441:
440:
416:Amorphous solid
410:
400:
399:
398:
377:
359:
349:
348:
347:
336:
334:Antiferromagnet
327:
325:Superparamagnet
318:
305:
304:Magnetic phases
297:
296:
295:
275:
267:
266:
265:
195:
187:
186:
185:
171:Order parameter
165:
164:Phase phenomena
157:
156:
155:
85:
75:
28:
23:
22:
18:Granular matter
15:
12:
11:
5:
7421:
7411:
7410:
7405:
7391:
7390:
7383:
7369:
7337:J. Fluid Mech.
7331:
7324:
7323:External links
7321:
7319:
7318:
7284:
7226:
7181:(25): 258003.
7165:
7106:
7086:
7066:
7016:
6981:
6939:
6919:
6901:
6854:
6807:
6752:
6739:
6719:
6697:
6695:
6692:
6691:
6690:
6685:
6679:
6674:
6669:
6664:
6659:
6654:
6652:Fragile matter
6649:
6642:
6639:
6618:
6615:
6602:
6599:
6596:
6593:
6590:
6577:
6574:
6553:Main article:
6550:
6547:
6522:
6519:
6507:
6506:
6492:
6484:
6471:
6468:
6446:
6442:
6419:
6415:
6394:
6391:
6371:
6368:
6344:
6322:
6318:
6314:
6311:
6308:
6305:
6302:
6282:
6260:
6256:
6235:
6215:
6195:
6175:
6172:
6169:
6166:
6146:
6143:
6138:
6135:
6131:
6110:
6107:
6102:
6099:
6095:
6074:
6054:
6030:
6005:
6002:
5990:
5987:
5984:
5981:
5978:
5971:
5968:
5965:
5962:
5958:
5952:
5947:
5942:
5937:
5930:
5926:
5920:
5917:
5912:
5909:
5905:
5901:
5897:
5888:
5882:
5879:
5874:
5871:
5867:
5863:
5859:
5853:
5850:
5847:
5844:
5839:
5835:
5829:
5826:
5821:
5818:
5814:
5810:
5806:
5800:
5795:
5791:
5784:
5781:
5775:
5772:
5769:
5764:
5761:
5758:
5754:
5750:
5743:
5740:
5735:
5731:
5724:
5721:
5713:
5708:
5704:
5673:
5670:
5665:
5661:
5654:
5651:
5645:
5642:
5639:
5636:
5633:
5607:
5604:
5601:
5598:
5594:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5559:
5555:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5523:
5520:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5493:
5489:
5483:
5478:
5474:
5470:
5467:
5464:
5461:
5458:
5455:
5433:
5430:
5427:
5424:
5404:
5401:
5398:
5395:
5373:
5370:
5364:
5361:
5358:
5355:
5352:
5349:
5344:
5340:
5336:
5330:
5325:
5320:
5317:
5314:
5309:
5305:
5301:
5298:
5295:
5291:
5287:
5282:
5277:
5274:
5271:
5266:
5262:
5258:
5255:
5252:
5248:
5244:
5233:
5228:
5224:
5220:
5217:
5214:
5211:
5207:
5203:
5200:
5197:
5193:
5189:
5185:
5181:
5178:
5175:
5172:
5169:
5165:
5161:
5157:
5153:
5150:
5147:
5144:
5141:
5138:
5134:
5130:
5108:
5102:
5098:
5095:
5092:
5087:
5083:
5079:
5076:
5073:
5070:
5065:
5061:
5056:
5052:
5049:
5046:
5042:
5038:
5034:
5031:
5028:
5025:
5021:
5016:
5013:
5010:
5005:
5001:
4997:
4994:
4990:
4986:
4981:
4977:
4974:
4971:
4968:
4965:
4961:
4957:
4953:
4947:
4943:
4940:
4937:
4934:
4930:
4926:
4923:
4920:
4916:
4912:
4888:
4883:
4880:
4877:
4872:
4868:
4865:
4862:
4857:
4853:
4849:
4846:
4843:
4840:
4835:
4831:
4826:
4822:
4819:
4816:
4812:
4808:
4807:
4804:
4801:
4798:
4795:
4792:
4788:
4785:
4782:
4777:
4773:
4769:
4766:
4762:
4758:
4757:
4755:
4750:
4745:
4742:
4739:
4736:
4733:
4730:
4725:
4721:
4717:
4714:
4710:
4684:
4679:
4675:
4671:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4644:
4641:
4638:
4634:
4628:
4624:
4618:
4613:
4609:
4603:
4598:
4594:
4591:
4587:
4582:
4573:
4569:
4565:
4560:
4555:
4551:
4521:
4518:
4515:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4485:
4482:
4479:
4475:
4471:
4466:
4461:
4457:
4453:
4450:
4443:
4440:
4435:
4432:
4408:
4405:
4402:
4399:
4396:
4393:
4370:
4367:
4364:
4361:
4358:
4355:
4350:
4347:
4344:
4340:
4334:
4329:
4325:
4321:
4317:
4312:
4309:
4306:
4302:
4298:
4294:
4291:
4288:
4285:
4282:
4254:
4249:
4246:
4241:
4237:
4232:
4226:
4221:
4217:
4214:
4211:
4208:
4204:
4199:
4196:
4192:
4188:
4185:
4182:
4177:
4173:
4168:
4163:
4159:
4154:
4149:
4146:
4143:
4138:
4135:
4131:
4126:
4122:
4118:
4091:
4086:
4083:
4080:
4075:
4072:
4068:
4063:
4059:
4055:
4051:
4048:
4045:
4038:
4035:
4029:
4024:
4020:
4016:
4012:
3982:
3977:
3974:
3971:
3963:
3959:
3956:
3949:
3945:
3940:
3936:
3932:
3925:
3922:
3916:
3913:
3906:
3903:
3897:
3893:
3890:
3887:
3882:
3878:
3873:
3869:
3865:
3861:
3858:
3855:
3852:
3849:
3846:
3841:
3837:
3832:
3822:
3819:
3816:
3813:
3809:
3805:
3802:
3799:
3792:
3789:
3783:
3778:
3774:
3770:
3766:
3733:
3727:
3722:
3718:
3715:
3712:
3709:
3705:
3700:
3697:
3693:
3689:
3686:
3683:
3678:
3674:
3669:
3665:
3661:
3654:
3651:
3645:
3642:
3639:
3636:
3633:
3629:
3625:
3622:
3619:
3614:
3610:
3605:
3600:
3596:
3593:
3590:
3587:
3584:
3580:
3576:
3573:
3571:
3569:
3565:
3559:
3554:
3550:
3546:
3541:
3537:
3531:
3527:
3523:
3520:
3515:
3510:
3506:
3501:
3496:
3491:
3487:
3483:
3479:
3476:
3473:
3470:
3467:
3463:
3459:
3456:
3453:
3448:
3444:
3439:
3434:
3430:
3427:
3424:
3421:
3418:
3414:
3410:
3407:
3405:
3402:
3398:
3395:
3392:
3389:
3386:
3383:
3378:
3374:
3369:
3365:
3364:
3335:
3331:
3328:
3325:
3322:
3318:
3314:
3311:
3309:
3307:
3303:
3298:
3294:
3291:
3288:
3285:
3281:
3277:
3273:
3269:
3266:
3263:
3260:
3256:
3251:
3244:
3241:
3235:
3232:
3229:
3226:
3223:
3219:
3215:
3212:
3209:
3206:
3202:
3197:
3193:
3190:
3187:
3184:
3181:
3177:
3173:
3170:
3168:
3166:
3162:
3157:
3152:
3148:
3144:
3140:
3136:
3131:
3127:
3123:
3118:
3113:
3110:
3107:
3103:
3100:
3097:
3094:
3091:
3087:
3083:
3080:
3077:
3074:
3070:
3065:
3061:
3058:
3055:
3052:
3049:
3045:
3041:
3038:
3036:
3033:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3007:
3003:
3002:
2981:
2977:
2974:
2971:
2967:
2946:
2926:
2904:
2899:
2896:
2893:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2846:
2843:
2840:
2835:
2831:
2827:
2824:
2821:
2818:
2813:
2809:
2804:
2800:
2797:
2796:
2793:
2790:
2787:
2784:
2781:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2739:
2736:
2733:
2728:
2724:
2720:
2719:
2717:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2689:
2685:
2656:
2650:
2646:
2642:
2637:
2633:
2628:
2623:
2619:
2616:
2613:
2609:
2587:
2581:
2577:
2573:
2568:
2564:
2559:
2555:
2534:
2530:
2527:
2524:
2520:
2516:
2513:
2491:
2487:
2483:
2478:
2474:
2451:
2447:
2443:
2438:
2434:
2424:with energies
2413:
2410:
2407:
2385:
2381:
2371:having energy
2360:
2340:
2328:
2325:
2306:
2305:Granular gases
2303:
2291:
2288:
2285:
2282:
2279:
2271:
2267:
2260:
2255:
2251:
2240:
2236:
2229:
2224:
2220:
2210:
2203:
2198:
2194:
2186:
2181:
2177:
2148:
2144:
2140:
2135:
2131:
2108:
2104:
2100:
2097:
2077:
2074:
2071:
2051:
2029:
2025:
2021:
2018:
1996:
1992:
1988:
1985:
1982:
1979:
1957:
1953:
1930:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1879:
1875:
1869:
1865:
1857:
1853:
1847:
1843:
1836:
1829:
1825:
1819:
1815:
1787:
1784:
1779:
1775:
1771:
1766:
1762:
1752:and therefore
1741:
1738:
1733:
1729:
1725:
1720:
1716:
1690:
1686:
1663:
1659:
1638:
1618:
1611:
1607:
1601:
1597:
1591:
1588:
1585:
1582:
1579:
1557:
1553:
1549:
1544:
1540:
1516:
1513:
1496:
1493:
1490:
1470:
1447:
1444:
1441:
1437:
1432:
1429:
1409:
1406:
1403:
1399:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1349:
1319:
1316:
1312:
1306:
1303:
1299:
1293:
1290:
1263:
1260:
1256:
1250:
1247:
1243:
1237:
1234:
1212:
1209:
1205:
1182:
1179:
1175:
1149:
1146:
1142:
1121:
1120:Janssen Effect
1118:
1078:
1074:
1070:
1065:
1061:
1057:
1054:
1051:
1029:
1025:
999:
995:
970:
967:
965:
962:
951:granular solid
920:
917:
909:complex system
823:pharmaceutical
764:asteroid belts
726:
725:
723:
722:
715:
708:
700:
697:
696:
695:
694:
681:
664:
663:
659:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
452:
451:
448:
447:
444:
443:
439:
438:
433:
431:Liquid crystal
428:
423:
418:
412:
411:
406:
405:
402:
401:
397:
396:
391:
386:
381:
372:
367:
361:
360:
357:Quasiparticles
355:
354:
351:
350:
346:
345:
340:
331:
322:
316:Superdiamagnet
313:
307:
306:
303:
302:
299:
298:
294:
293:
288:
283:
277:
276:
273:
272:
269:
268:
264:
263:
258:
253:
248:
243:
241:Thermoelectric
238:
236:Superconductor
233:
228:
223:
218:
216:Mott insulator
213:
208:
203:
197:
196:
193:
192:
189:
188:
184:
183:
178:
173:
167:
166:
163:
162:
159:
158:
154:
153:
148:
143:
138:
133:
128:
123:
118:
113:
108:
103:
98:
93:
87:
86:
81:
80:
77:
76:
74:
73:
68:
63:
57:
54:
53:
45:
44:
26:
9:
6:
4:
3:
2:
7420:
7409:
7406:
7404:
7401:
7400:
7398:
7388:
7384:
7382:
7378:
7374:
7370:
7366:
7362:
7358:
7354:
7350:
7346:
7342:
7339:
7338:
7332:
7330:
7327:
7326:
7314:
7310:
7306:
7302:
7295:
7288:
7280:
7276:
7272:
7268:
7264:
7260:
7256:
7252:
7248:
7244:
7237:
7230:
7222:
7218:
7214:
7210:
7206:
7202:
7198:
7194:
7189:
7184:
7180:
7176:
7169:
7161:
7157:
7152:
7147:
7143:
7139:
7134:
7129:
7126:(5): 055701.
7125:
7121:
7117:
7110:
7103:
7099:
7095:
7090:
7083:
7079:
7076:
7070:
7062:
7058:
7054:
7050:
7045:
7040:
7036:
7032:
7025:
7023:
7021:
7012:
7008:
7004:
7000:
6996:
6992:
6985:
6977:
6973:
6969:
6965:
6961:
6957:
6950:
6943:
6937:
6933:
6929:
6923:
6915:
6908:
6906:
6897:
6893:
6889:
6885:
6881:
6877:
6874:(1): 012902.
6873:
6869:
6865:
6858:
6850:
6846:
6842:
6838:
6834:
6830:
6826:
6822:
6818:
6811:
6803:
6799:
6795:
6791:
6787:
6783:
6779:
6775:
6771:
6767:
6763:
6756:
6749:
6743:
6737:
6736:0-471-92422-9
6733:
6729:
6723:
6716:
6715:0-387-98656-1
6712:
6708:
6702:
6698:
6689:
6686:
6683:
6682:ÎĽ(I) rheology
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6644:
6638:
6636:
6632:
6628:
6624:
6614:
6600:
6597:
6594:
6591:
6588:
6573:
6572:
6568:
6565:
6561:
6556:
6546:
6544:
6540:
6536:
6535:Squeaky Beach
6527:
6518:
6516:
6512:
6505:
6501:
6497:
6493:
6490:
6485:
6481:
6477:
6476:
6475:
6467:
6465:
6460:
6444:
6440:
6417:
6413:
6392:
6369:
6358:
6342:
6320:
6316:
6312:
6309:
6306:
6303:
6280:
6258:
6254:
6233:
6213:
6193:
6170:
6141:
6136:
6133:
6129:
6108:
6105:
6100:
6097:
6093:
6052:
6044:
6028:
6019:
6010:
6001:
5985:
5979:
5976:
5969:
5966:
5963:
5960:
5956:
5950:
5940:
5928:
5924:
5918:
5915:
5910:
5907:
5903:
5899:
5895:
5886:
5880:
5877:
5872:
5869:
5865:
5861:
5857:
5851:
5848:
5845:
5842:
5837:
5833:
5827:
5824:
5819:
5816:
5812:
5808:
5804:
5793:
5789:
5782:
5779:
5773:
5770:
5767:
5762:
5759:
5756:
5752:
5748:
5741:
5738:
5733:
5729:
5722:
5719:
5706:
5702:
5693:
5691:
5671:
5668:
5663:
5659:
5652:
5649:
5643:
5637:
5631:
5622:
5605:
5602:
5599:
5596:
5592:
5586:
5580:
5574:
5565:
5557:
5553:
5549:
5543:
5537:
5534:
5528:
5521:
5518:
5514:
5508:
5505:
5499:
5491:
5487:
5481:
5476:
5472:
5468:
5462:
5456:
5453:
5445:
5431:
5428:
5425:
5422:
5399:
5393:
5384:
5371:
5368:
5359:
5356:
5353:
5350:
5342:
5338:
5334:
5328:
5323:
5315:
5307:
5303:
5299:
5296:
5293:
5289:
5285:
5280:
5272:
5264:
5260:
5256:
5253:
5250:
5246:
5242:
5231:
5226:
5222:
5218:
5215:
5209:
5205:
5201:
5198:
5195:
5191:
5187:
5183:
5179:
5176:
5170:
5167:
5163:
5159:
5155:
5151:
5148:
5145:
5142:
5139:
5136:
5132:
5128:
5120:
5106:
5100:
5093:
5085:
5081:
5077:
5071:
5063:
5059:
5054:
5050:
5047:
5044:
5040:
5036:
5032:
5029:
5023:
5019:
5011:
5003:
4999:
4995:
4992:
4988:
4984:
4979:
4975:
4972:
4966:
4963:
4959:
4955:
4951:
4945:
4941:
4938:
4935:
4932:
4928:
4924:
4921:
4918:
4914:
4910:
4901:
4881:
4870:
4863:
4855:
4851:
4847:
4841:
4833:
4829:
4824:
4820:
4817:
4814:
4810:
4802:
4796:
4793:
4783:
4775:
4771:
4767:
4764:
4760:
4753:
4748:
4740:
4737:
4734:
4731:
4723:
4719:
4715:
4712:
4708:
4699:
4696:
4682:
4677:
4673:
4669:
4665:
4662:
4659:
4653:
4647:
4642:
4639:
4636:
4632:
4626:
4622:
4611:
4607:
4601:
4596:
4592:
4589:
4585:
4580:
4571:
4567:
4563:
4558:
4553:
4549:
4536:
4533:
4519:
4516:
4513:
4509:
4506:
4503:
4500:
4494:
4488:
4483:
4480:
4477:
4473:
4469:
4459:
4455:
4451:
4448:
4441:
4438:
4433:
4430:
4406:
4403:
4397:
4391:
4382:
4368:
4365:
4359:
4353:
4348:
4345:
4342:
4338:
4327:
4323:
4319:
4315:
4310:
4307:
4304:
4300:
4296:
4292:
4286:
4280:
4272:
4267:
4252:
4247:
4239:
4235:
4230:
4224:
4219:
4212:
4206:
4202:
4197:
4194:
4190:
4183:
4175:
4171:
4166:
4161:
4157:
4152:
4147:
4144:
4141:
4136:
4133:
4129:
4124:
4120:
4116:
4107:
4104:
4089:
4084:
4081:
4078:
4073:
4070:
4066:
4061:
4057:
4053:
4046:
4043:
4036:
4033:
4027:
4022:
4018:
4014:
4010:
3998:
3995:
3980:
3975:
3972:
3969:
3961:
3954:
3947:
3943:
3938:
3934:
3930:
3923:
3914:
3911:
3904:
3901:
3895:
3888:
3880:
3876:
3871:
3867:
3863:
3856:
3853:
3850:
3847:
3839:
3835:
3830:
3820:
3814:
3811:
3807:
3803:
3800:
3797:
3790:
3787:
3781:
3776:
3772:
3768:
3764:
3752:
3749:
3731:
3725:
3720:
3713:
3707:
3703:
3698:
3695:
3691:
3684:
3676:
3672:
3667:
3663:
3659:
3652:
3649:
3643:
3640:
3637:
3631:
3627:
3620:
3612:
3608:
3603:
3598:
3594:
3591:
3585:
3582:
3578:
3574:
3572:
3563:
3557:
3552:
3548:
3544:
3539:
3535:
3529:
3525:
3521:
3518:
3513:
3508:
3504:
3499:
3494:
3489:
3485:
3481:
3477:
3474:
3471:
3465:
3461:
3454:
3446:
3442:
3437:
3432:
3428:
3425:
3419:
3416:
3412:
3408:
3406:
3400:
3393:
3390:
3387:
3384:
3376:
3372:
3367:
3354:
3351:
3333:
3326:
3320:
3316:
3312:
3310:
3301:
3296:
3289:
3283:
3279:
3275:
3271:
3264:
3258:
3254:
3249:
3242:
3239:
3233:
3230:
3227:
3221:
3217:
3210:
3204:
3200:
3195:
3191:
3188:
3182:
3179:
3175:
3171:
3169:
3160:
3155:
3150:
3146:
3142:
3138:
3134:
3129:
3125:
3121:
3116:
3111:
3108:
3105:
3101:
3098:
3095:
3089:
3085:
3078:
3072:
3068:
3063:
3059:
3056:
3050:
3047:
3043:
3039:
3037:
3031:
3024:
3021:
3018:
3015:
3009:
3005:
2979:
2975:
2972:
2969:
2965:
2944:
2897:
2894:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2848:
2841:
2833:
2829:
2825:
2819:
2811:
2807:
2802:
2798:
2791:
2788:
2782:
2779:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2734:
2726:
2722:
2715:
2710:
2704:
2701:
2698:
2695:
2687:
2683:
2674:
2669:
2654:
2648:
2644:
2640:
2635:
2631:
2626:
2621:
2617:
2614:
2611:
2607:
2585:
2579:
2575:
2571:
2566:
2562:
2557:
2553:
2532:
2528:
2525:
2522:
2518:
2514:
2511:
2489:
2485:
2481:
2476:
2472:
2449:
2445:
2441:
2436:
2432:
2411:
2408:
2405:
2383:
2379:
2358:
2338:
2324:
2322:
2317:
2313:
2302:
2289:
2286:
2283:
2280:
2277:
2269:
2265:
2258:
2253:
2249:
2238:
2234:
2227:
2222:
2218:
2208:
2201:
2196:
2184:
2179:
2162:
2146:
2138:
2133:
2106:
2102:
2098:
2095:
2075:
2072:
2069:
2049:
2027:
2023:
2019:
2016:
1994:
1990:
1986:
1983:
1980:
1977:
1955:
1951:
1942:
1928:
1919:
1903:
1900:
1897:
1891:
1888:
1885:
1877:
1873:
1867:
1863:
1855:
1851:
1845:
1841:
1834:
1827:
1823:
1817:
1813:
1802:
1799:
1785:
1782:
1777:
1773:
1769:
1764:
1760:
1739:
1736:
1731:
1727:
1723:
1718:
1714:
1704:
1688:
1684:
1661:
1657:
1636:
1609:
1605:
1599:
1595:
1586:
1583:
1580:
1577:
1555:
1551:
1547:
1542:
1538:
1529:
1524:
1522:
1512:
1508:
1494:
1491:
1488:
1468:
1445:
1442:
1439:
1435:
1430:
1427:
1401:
1397:
1393:
1390:
1384:
1381:
1378:
1375:
1363:
1359:
1353:
1347:
1338:
1335:
1317:
1314:
1310:
1304:
1301:
1297:
1291:
1288:
1279:
1261:
1258:
1254:
1248:
1245:
1241:
1235:
1232:
1210:
1207:
1203:
1180:
1177:
1173:
1163:
1147:
1144:
1140:
1130:
1128:
1117:
1114:
1110:
1106:
1102:
1098:
1094:
1076:
1072:
1068:
1063:
1059:
1055:
1052:
1027:
1023:
1015:
997:
993:
983:
975:
961:
959:
954:
952:
947:
945:
940:
938:
934:
930:
926:
916:
914:
913:Magnus effect
910:
904:
902:
898:
894:
890:
885:
883:
879:
875:
871:
870:
865:
861:
857:
852:
850:
846:
842:
838:
834:
832:
828:
824:
820:
816:
812:
811:bearing balls
808:
804:
800:
796:
792:
788:
784:
780:
775:
773:
769:
765:
761:
757:
752:
748:
744:
741:
737:
733:
721:
716:
714:
709:
707:
702:
701:
699:
698:
692:
682:
679:
674:
668:
667:
666:
665:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
456:Van der Waals
454:
453:
446:
445:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
413:
409:
404:
403:
395:
392:
390:
387:
385:
382:
380:
376:
373:
371:
368:
366:
363:
362:
358:
353:
352:
344:
341:
339:
335:
332:
330:
326:
323:
321:
317:
314:
312:
309:
308:
301:
300:
292:
289:
287:
284:
282:
279:
278:
271:
270:
262:
259:
257:
254:
252:
251:Ferroelectric
249:
247:
246:Piezoelectric
244:
242:
239:
237:
234:
232:
229:
227:
224:
222:
221:Semiconductor
219:
217:
214:
212:
209:
207:
204:
202:
199:
198:
191:
190:
182:
179:
177:
174:
172:
169:
168:
161:
160:
152:
149:
147:
144:
142:
141:Superfluidity
139:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
88:
84:
79:
78:
72:
69:
67:
64:
62:
59:
58:
56:
55:
51:
47:
46:
43:
40:
39:
32:
19:
7371:Mester, L.,
7340:
7335:
7304:
7300:
7287:
7246:
7242:
7229:
7178:
7174:
7168:
7123:
7119:
7109:
7101:
7089:
7069:
7034:
7030:
6994:
6990:
6984:
6962:(18): 8212.
6959:
6955:
6942:
6927:
6922:
6913:
6871:
6867:
6857:
6824:
6820:
6810:
6772:(2): 121–8.
6769:
6765:
6755:
6747:
6742:
6727:
6722:
6706:
6701:
6672:Particulates
6667:Metal powder
6622:
6620:
6579:
6570:
6559:
6558:
6543:silo honking
6532:
6515:event-driven
6511:time-stepped
6508:
6473:
6461:
6015:
5694:
5623:
5446:
5386:Solving for
5385:
5121:
4902:
4700:
4697:
4537:
4534:
4383:
4268:
4108:
4105:
3999:
3996:
3753:
3750:
3355:
3352:
2670:
2330:
2308:
2163:
1943:
1920:
1803:
1800:
1705:
1525:
1518:
1509:
1339:
1336:
1280:
1164:
1131:
1123:
1113:shear stress
1101:force chains
1097:force chains
980:
957:
955:
950:
948:
944:granular gas
943:
941:
922:
905:
886:
867:
853:
835:
776:
768:Solar System
731:
729:
586:von Klitzing
425:
291:Kondo effect
151:Time crystal
131:Fermi liquid
7343:: 347–372.
6705:Duran, J.,
6560:Granulation
6555:Granulation
6549:Granulation
2316:dissipative
1111:. When the
933:Dissipation
919:Definitions
827:agriculture
803:corn flakes
740:macroscopic
408:Soft matter
329:Ferromagnet
7397:Categories
7188:2008.01920
7133:1705.02984
7044:1805.07865
6694:References
6529:Sand dunes
6504:sandsheets
2327:Ulam Model
1093:hysteresis
929:rigid body
862:Brigadier
825:industry,
551:Louis NĂ©el
541:Schrieffer
449:Scientists
343:Spin glass
338:Metamagnet
320:Paramagnet
136:Supersolid
7307:: 52–61.
7271:1944-8007
7221:220968720
7061:119084790
6849:214483792
6841:0022-1120
6635:level-set
6631:presented
6598:±
6589:ϕ
6571:granules.
6567:particles
6489:oscillons
6441:ϕ
6414:ϕ
6393:ϕ
6390:Δ
6370:ϕ
6367:Δ
6317:ϕ
6313:−
6310:ϕ
6307:≡
6304:ϕ
6301:Δ
6255:ϕ
6194:ϕ
6171:ϕ
6165:Σ
6145:Σ
6142:−
6134:−
6130:ϕ
6106:−
6098:−
6094:ϕ
6073:Σ
6053:ϕ
5986:λ
5961:λ
5946:∞
5929:ε
5908:λ
5900:−
5870:λ
5852:−
5846:ε
5838:ε
5817:λ
5809:−
5799:∞
5790:∫
5771:ε
5763:ε
5760:λ
5757:−
5749:⋅
5739:ε
5734:−
5712:∞
5703:∫
5669:ε
5664:−
5638:ε
5632:ρ
5597:λ
5581:λ
5572:⇒
5566:λ
5544:λ
5529:λ
5515:λ
5512:⇒
5509:δ
5500:δ
5482:λ
5473:∫
5463:λ
5454:λ
5429:λ
5423:δ
5400:λ
5351:λ
5329:⏟
5304:ε
5297:λ
5294:−
5261:ε
5254:λ
5251:−
5223:∫
5213:Γ
5196:λ
5174:Γ
5171:−
5137:λ
5082:ε
5060:ε
5048:λ
5045:−
5027:Γ
5000:ε
4996:λ
4993:−
4970:Γ
4967:−
4925:ε
4922:λ
4919:−
4879:Γ
4852:ε
4830:ε
4818:λ
4815:−
4800:Γ
4797:−
4772:ε
4768:λ
4765:−
4720:ε
4716:λ
4713:−
4674:ε
4663:ε
4654:ε
4648:ρ
4643:ε
4640:λ
4637:−
4623:ε
4617:∞
4608:∫
4590:−
4568:λ
4517:ε
4510:−
4504:ε
4495:ε
4489:ρ
4484:ε
4481:λ
4478:−
4470:ε
4465:∞
4456:∫
4452:−
4442:λ
4369:ε
4360:ε
4354:ρ
4349:ε
4346:λ
4343:−
4333:∞
4324:∫
4311:ε
4308:λ
4305:−
4287:λ
4245:Γ
4240:−
4207:ε
4195:−
4172:ε
4145:ε
4134:−
4121:ε
4082:ε
4058:ε
4050:⇒
4019:ε
3973:ε
3958:Γ
3935:ε
3921:Γ
3915:−
3877:ε
3868:−
3836:ε
3818:→
3773:ε
3708:ε
3673:ε
3644:⋅
3635:Γ
3609:ε
3589:Γ
3586:−
3549:ε
3536:ε
3526:ε
3505:ε
3478:⋅
3469:Γ
3443:ε
3423:Γ
3420:−
3373:ε
3321:ε
3284:ε
3259:ε
3234:⋅
3225:Γ
3205:ε
3186:Γ
3183:−
3147:ε
3126:ε
3102:⋅
3093:Γ
3073:ε
3054:Γ
3051:−
3010:ε
2925:Γ
2892:Γ
2830:ε
2808:ε
2786:Γ
2783:−
2723:ε
2684:ε
2675:equation:
2645:ε
2632:ε
2615:−
2576:ε
2563:ε
2515:∈
2486:ε
2473:ε
2446:ε
2433:ε
2380:ε
2331:Consider
2290:β
2287:
2281:−
2266:ℓ
2259:˙
2250:ε
2235:ℓ
2228:˙
2219:ε
2202:˙
2193:Δ
2185:˙
2176:Δ
2143:Δ
2130:Δ
2107:μ
2103:θ
2099:≥
2096:θ
2076:β
2070:α
2050:θ
2028:μ
2024:θ
2020:≤
2017:θ
1991:ϕ
1978:μ
1956:μ
1952:θ
1929:θ
1904:β
1898:θ
1892:
1874:ℓ
1864:σ
1852:ℓ
1842:σ
1728:σ
1715:σ
1685:σ
1658:σ
1637:β
1606:ℓ
1596:ℓ
1587:
1578:α
1552:ℓ
1539:ℓ
1521:dilatancy
1443:μ
1428:λ
1402:λ
1391:−
1385:
1379:−
1368:∞
1311:σ
1298:σ
1289:μ
1255:σ
1242:σ
1225:, where
1204:σ
1174:σ
1141:σ
1073:θ
1069:−
1060:θ
1053:θ
1050:Δ
1024:θ
994:θ
860:physicist
845:suspended
772:asteroids
743:particles
651:Wetterich
631:Abrikosov
546:Josephson
516:Van Vleck
506:Luttinger
379:Polariton
311:Diamagnet
231:Conductor
226:Semimetal
211:Insulator
126:Fermi gas
7365:30744277
7279:44736386
7213:33416399
7160:29481202
7078:Archived
7011:10034316
6896:73456295
6888:30780222
6802:25375365
6794:15689950
6641:See also
5522:′
5324:⟩
5286:⟨
5281:⟩
5243:⟨
5107:⟩
5037:⟨
5020:⟩
4985:⟨
4952:⟩
4911:⟨
4683:⟩
4670:⟨
4520:⟩
4514:⟨
4316:⟩
4297:⟨
4220:⟩
4203:⟨
4191:⟩
4167:⟨
4148:⟩
4142:⟨
4130:⟩
4117:⟨
4085:⟩
4079:⟨
4067:⟩
4054:⟨
4028:⟩
4015:⟨
3976:⟩
3970:⟨
3944:⟩
3931:⟨
3896:⟩
3872:⟨
3864:⟩
3831:⟨
3782:⟩
3769:⟨
3721:⟩
3704:⟨
3692:⟩
3668:⟨
3628:⟩
3604:⟨
3564:⟩
3500:⟨
3495:⟩
3482:⟨
3462:⟩
3438:⟨
3401:⟩
3368:⟨
3334:⟩
3317:⟨
3297:⟩
3280:⟨
3272:⟩
3255:⟨
3218:⟩
3201:⟨
3156:⟩
3143:⟨
3135:⟩
3122:⟨
3112:⟩
3106:⟨
3086:⟩
3069:⟨
3032:⟩
3006:⟨
1105:vaulting
841:cohesive
817:, whose
760:icebergs
747:friction
691:Category
636:Ginzburg
611:Laughlin
571:Kadanoff
526:Shockley
511:Anderson
466:von Laue
116:Bose gas
7345:Bibcode
7251:Bibcode
7193:Bibcode
7138:Bibcode
6964:Bibcode
6827:: A23.
6774:Bibcode
6539:booming
6496:ripples
6018:jamming
1109:arching
982:Coulomb
897:liquids
856:soldier
837:Powders
766:of the
762:and to
641:Leggett
616:Störmer
601:Bednorz
561:Giaever
531:Bardeen
521:Hubbard
496:Peierls
486:Onsager
436:Polymer
421:Colloid
384:Polaron
375:Plasmon
370:Exciton
7379:
7363:
7277:
7269:
7219:
7211:
7158:
7059:
7009:
6934:
6894:
6886:
6847:
6839:
6800:
6792:
6734:
6713:
6564:powder
6502:, and
4419:, and
4384:Where
2917:where
1921:where
1584:arctan
1420:where
1127:Stevin
893:solids
878:nature
829:, and
809:, and
799:coffee
751:grains
689:
656:Perdew
646:Parisi
606:MĂĽller
596:Rohrer
591:Binnig
581:Wilson
576:Fisher
536:Cooper
501:Landau
389:Magnon
365:Phonon
206:Plasma
106:Plasma
96:Liquid
61:Phases
7361:S2CID
7297:(PDF)
7275:S2CID
7239:(PDF)
7217:S2CID
7183:arXiv
7128:arXiv
7057:S2CID
7039:arXiv
7037:(2).
6952:(PDF)
6892:S2CID
6845:S2CID
6798:S2CID
6601:0.001
6595:0.646
6500:dunes
4698:now:
2009:, so
901:gases
899:, or
882:water
847:in a
749:when
736:solid
556:Esaki
481:Bloch
476:Debye
471:Bragg
461:Onnes
394:Roton
91:Solid
7377:ISBN
7267:ISSN
7209:PMID
7156:PMID
7007:PMID
6932:ISBN
6884:PMID
6837:ISSN
6790:PMID
6732:ISBN
6711:ISBN
6513:and
6483:top.
6462:The
2671:The
1461:and
1107:and
884:)".
854:The
807:salt
795:rice
791:sand
787:coal
783:nuts
779:snow
626:Tsui
621:Yang
566:Kohn
491:Mott
7353:doi
7341:587
7309:doi
7305:100
7259:doi
7201:doi
7179:125
7146:doi
7124:120
7100:",
7096:, "
7049:doi
6999:doi
6972:doi
6960:117
6876:doi
6829:doi
6825:886
6782:doi
2284:tan
1889:tan
1382:exp
925:DOF
849:gas
181:QCP
101:Gas
71:QCP
7399::
7359:.
7351:.
7303:.
7299:.
7273:.
7265:.
7257:.
7247:44
7245:.
7241:.
7215:.
7207:.
7199:.
7191:.
7177:.
7154:.
7144:.
7136:.
7122:.
7118:.
7055:.
7047:.
7035:21
7033:.
7019:^
7005:.
6995:58
6993:.
6970:.
6958:.
6954:.
6904:^
6890:.
6882:.
6872:99
6870:.
6866:.
6843:.
6835:.
6823:.
6819:.
6796:.
6788:.
6780:.
6768:.
6764:.
6545:.
6498:,
6045:,
6041:,
5444::
4381:.
4273::
2668:.
2254:11
2223:22
1868:22
1846:11
1828:22
1818:11
1798:.
1778:21
1765:12
1732:21
1719:12
1689:22
1662:11
1507:.
960:.
915:.
895:,
851:.
833:.
805:,
801:,
797:,
793:,
789:,
785:,
781:,
774:.
756:ÎĽm
738:,
730:A
7367:.
7355::
7347::
7315:.
7311::
7281:.
7261::
7253::
7223:.
7203::
7195::
7185::
7162:.
7148::
7140::
7130::
7063:.
7051::
7041::
7013:.
7001::
6978:.
6974::
6966::
6898:.
6878::
6851:.
6831::
6804:.
6784::
6776::
6770:4
6717:.
6592:=
6445:c
6418:c
6343:J
6321:c
6281:J
6259:c
6234:J
6214:J
6174:)
6168:(
6137:1
6109:T
6101:1
6029:T
5989:)
5983:(
5980:g
5977:=
5970:1
5967:+
5964:T
5957:1
5951:=
5941:0
5936:|
5925:)
5919:T
5916:1
5911:+
5904:(
5896:e
5887:)
5881:T
5878:1
5873:+
5866:(
5862:T
5858:1
5849:=
5843:d
5834:)
5828:T
5825:1
5820:+
5813:(
5805:e
5794:0
5783:T
5780:1
5774:=
5768:d
5753:e
5742:T
5730:e
5723:T
5720:1
5707:0
5688:(
5672:T
5660:e
5653:T
5650:1
5644:=
5641:)
5635:(
5606:1
5603:+
5600:T
5593:1
5587:=
5584:)
5578:(
5575:g
5569:)
5563:(
5558:2
5554:g
5550:=
5547:)
5541:(
5538:g
5535:+
5532:)
5526:(
5519:g
5506:d
5503:)
5497:(
5492:2
5488:g
5477:0
5469:=
5466:)
5460:(
5457:g
5432:z
5426:=
5403:)
5397:(
5394:g
5372:z
5369:d
5363:)
5360:t
5357:,
5354:z
5348:(
5343:2
5339:g
5335:=
5319:)
5316:t
5313:(
5308:j
5300:z
5290:e
5276:)
5273:t
5270:(
5265:i
5257:z
5247:e
5232:1
5227:0
5219:t
5216:d
5210:+
5206:)
5202:t
5199:,
5192:(
5188:g
5184:)
5180:t
5177:d
5168:1
5164:(
5160:=
5156:)
5152:t
5149:d
5146:+
5143:t
5140:,
5133:(
5129:g
5101:)
5097:)
5094:t
5091:(
5086:j
5078:+
5075:)
5072:t
5069:(
5064:i
5055:(
5051:z
5041:e
5033:t
5030:d
5024:+
5015:)
5012:t
5009:(
5004:i
4989:e
4980:)
4976:t
4973:d
4964:1
4960:(
4956:=
4946:)
4942:t
4939:d
4936:+
4933:t
4929:(
4915:e
4882:t
4871:)
4867:)
4864:t
4861:(
4856:j
4848:+
4845:)
4842:t
4839:(
4834:i
4825:(
4821:z
4811:e
4803:t
4794:1
4787:)
4784:t
4781:(
4776:i
4761:e
4754:{
4749:=
4744:)
4741:t
4738:d
4735:+
4732:t
4729:(
4724:i
4709:e
4678:n
4666:=
4660:d
4657:)
4651:(
4633:e
4627:n
4612:0
4602:n
4597:)
4593:1
4586:(
4581:=
4572:n
4564:d
4559:g
4554:n
4550:d
4507:=
4501:d
4498:)
4492:(
4474:e
4460:0
4449:=
4439:d
4434:g
4431:d
4407:1
4404:=
4401:)
4398:0
4395:(
4392:g
4366:d
4363:)
4357:(
4339:e
4328:0
4320:=
4301:e
4293:=
4290:)
4284:(
4281:g
4253:t
4248:3
4236:e
4231:)
4225:2
4216:)
4213:0
4210:(
4198:2
4187:)
4184:0
4181:(
4176:2
4162:(
4158:=
4153:2
4137:2
4125:2
4090:2
4074:2
4071:=
4062:2
4047:0
4044:=
4037:t
4034:d
4023:2
4011:d
3981:2
3962:3
3955:2
3948:+
3939:2
3924:3
3912:=
3905:t
3902:d
3892:)
3889:t
3886:(
3881:2
3860:)
3857:t
3854:d
3851:+
3848:t
3845:(
3840:2
3821:0
3815:t
3812:d
3808:m
3804:i
3801:l
3798:=
3791:t
3788:d
3777:2
3765:d
3732:)
3726:2
3717:)
3714:t
3711:(
3699:2
3696:+
3688:)
3685:t
3682:(
3677:2
3664:2
3660:(
3653:3
3650:1
3641:t
3638:d
3632:+
3624:)
3621:t
3618:(
3613:2
3599:)
3595:t
3592:d
3583:1
3579:(
3575:=
3558:2
3553:j
3545:+
3540:j
3530:i
3522:2
3519:+
3514:2
3509:i
3490:2
3486:z
3475:t
3472:d
3466:+
3458:)
3455:t
3452:(
3447:2
3433:)
3429:t
3426:d
3417:1
3413:(
3409:=
3397:)
3394:t
3391:d
3388:+
3385:t
3382:(
3377:2
3330:)
3327:t
3324:(
3313:=
3302:)
3293:)
3290:t
3287:(
3276:+
3268:)
3265:t
3262:(
3250:(
3243:2
3240:1
3231:t
3228:d
3222:+
3214:)
3211:t
3208:(
3196:)
3192:t
3189:d
3180:1
3176:(
3172:=
3161:)
3151:j
3139:+
3130:i
3117:(
3109:z
3099:t
3096:d
3090:+
3082:)
3079:t
3076:(
3064:)
3060:t
3057:d
3048:1
3044:(
3040:=
3028:)
3025:t
3022:d
3019:+
3016:t
3013:(
2980:]
2976:1
2973:,
2970:0
2966:[
2945:z
2898:t
2895:d
2888::
2885:y
2882:t
2879:i
2876:l
2873:i
2870:b
2867:a
2864:b
2861:o
2858:r
2855:p
2849:)
2845:)
2842:t
2839:(
2834:j
2826:+
2823:)
2820:t
2817:(
2812:i
2803:(
2799:z
2792:t
2789:d
2780:1
2776::
2773:y
2770:t
2767:i
2764:l
2761:i
2758:b
2755:a
2752:b
2749:o
2746:r
2743:p
2738:)
2735:t
2732:(
2727:i
2716:{
2711:=
2708:)
2705:t
2702:d
2699:+
2696:t
2693:(
2688:i
2655:)
2649:j
2641:+
2636:i
2627:(
2622:)
2618:z
2612:1
2608:(
2586:)
2580:j
2572:+
2567:i
2558:(
2554:z
2533:]
2529:1
2526:,
2523:0
2519:[
2512:z
2490:j
2482:+
2477:i
2450:j
2442:,
2437:i
2412:j
2409:,
2406:i
2384:i
2359:i
2339:N
2278:=
2270:1
2239:2
2209:=
2197:1
2180:2
2147:2
2139:,
2134:1
2073:,
1995:u
1987:g
1984:t
1981:=
1907:)
1901:+
1895:(
1886:=
1878:1
1856:2
1835:=
1824:F
1814:F
1786:0
1783:=
1774:F
1770:=
1761:F
1740:0
1737:=
1724:=
1617:)
1610:2
1600:1
1590:(
1581:=
1556:2
1548:,
1543:1
1495:0
1492:=
1489:z
1469:R
1446:K
1440:2
1436:R
1431:=
1408:]
1405:)
1398:/
1394:z
1388:(
1376:1
1373:[
1364:p
1360:=
1357:)
1354:z
1351:(
1348:p
1318:r
1315:r
1305:z
1302:r
1292:=
1262:z
1259:z
1249:r
1246:r
1236:=
1233:K
1211:z
1208:z
1181:r
1178:r
1148:z
1145:z
1077:r
1064:m
1056:=
1028:r
998:m
858:/
719:e
712:t
705:v
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
