Scaling ansatz for the jamming transition

Abstract

We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.jamming transition | scaling ansatz | nonequilibrium critical phenomena T he existence of criticality at the jamming transition suggests that universal physics underlies rigidity in disordered solids ranging from glasses to granular materials (1). The jamming transition marks the onset of rigidity in athermal sphere packings and was originally proposed as a zero-temperature transition (2, 3) for soft repulsive spheres in a nonequilibrium "jamming phase diagram" (4) of varying packing density and applied shear. Many studies have documented behaviors characteristic of critical phenomena near the jamming transition, including power law scaling (2, 3, 5) and scaling collapses (6-13) of numerous properties, with the expression of quantities in terms of scaling functions, diverging length scales (6,(14)(15)(16)(17)(18)(19), and finite-size scaling (10,12,20). Theories have been developed to individually understand and relate some of these power laws (15,16,21,22), but a unified scaling theory has been lacking. Here, we develop such a theory by proposing a scaling ansatz for the jamming critical point in terms of the fields originally identified by the jamming phase diagram, namely density and shear.The critical point scaling ansatz introduced by Widom (23) in the 1960s was a key advance in the theory of critical phenomena that set the stage for the development of the renormalization group. The ansatz writes the state functions near continuous equilibrium phase transitions in terms of power law ratios of the control parameters. By positing a scaling function for the free energy, it exploits the fact that quantities, such as the specific heat, magnetization, and susceptibility, are derivatives of the free energy to derive relations not only among their scaling exponents but among their scaling functions. Thus, the scaling ansatz provides a unified and comprehensive description of systems exhibiting what later was realized to be an emergent scale invariance.Unlike most systems exhibiting a dynamical scale invariance, we find that a jammed system can also be viewed as a material with critical properties that are determined by a state function, analo...

Scaling Theory for the Frictionless Unjamming Transition

“…These singular distributions are in sharp contrast to the broadening of order parameter distributions approaching a thermal critical point. We will show that the power laws describing the evolution of global quantities approaching unjamming are a consequence of the singularities of p DO (α).In particular, the total number of contacts (N C ), with ∆N C = N C (E G ) − N C (0) scales as:a form observed in several studies of jamming [16,19,26,27,31]. The scaling of the total grain area A G , with…”

“…a form observed in several studies of jamming [16,19,26,27,31]. The scaling of the total grain area A G , with…”

“…The unjamming transition of soft spheres exhibits properties reminiscent of critical points in equilibrium systems. Observations include power laws [16], a scaling form for the energy analogous to free energy and resulting relationships between scaling exponents [26], scaling collapse of dynamical quantities such as viscosity [28], and indications of diverging length scales [21]. Many scaling properties of soft particles near the jamming transition have been analysed in detail [29,30], and finite-size scaling studies seem to suggest a mixed order transition with two critical exponents [18,21].…”

“…Fg, 81.05.Rm, Introduction: The jamming of soft particles has been used as a paradigmatic model of granular [1][2][3][4][5][6][7][8][9] and glassy systems [10][11][12], active matter [13] and biological tissues [14]. Frictionless soft disks and spheres serve as a first approximation to many theoretical models and have been extensively investigated over the last decade [15][16][17][18][19][20][21][22][23][24][25][26][27]. The unjamming transition of soft spheres exhibits properties reminiscent of critical points in equilibrium systems.…”

“…Frictionless soft disks and spheres serve as a first approximation to many theoretical models and have been extensively investigated over the last decade [15][16][17][18][19][20][21][22][23][24][25][26][27]. The unjamming transition of soft spheres exhibits properties reminiscent of critical points in equilibrium systems.…”