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Register a new userOut-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain
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Elisabeth Agoritsas
Publication date
2019
fields
Physics
and research's language is
English
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As an extension of the isotropic setting presented in the companion paper [J. Phys. A 52, 144002 (2019)], we consider the Langevin dynamics of a many-body system of pairwise interacting particles in $d$ dimensions, submitted to an external shear strain. We show that the anisotropy introduced by the shear strain can be simply addressed by moving into the co-shearing frame, leading to simple dynamical mean field equations in the limit ${dtoinfty}$. The dynamics is then controlled by a single one-dimensional effective stochastic process which depends on three distinct strain-dependent kernels - self-consistently determined by the process itself - encoding the effective restoring force, friction and noise terms due to the particle interactions. From there one can compute dynamical observables such as particle mean-square displacements and shear stress fluctuations, and eventually aim at providing an exact ${d to infty}$ benchmark for liquid and glass rheology. As an application of our results, we derive dynamically the state-following equations that describe the static response of a glass to a finite shear strain until it yields.
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We consider the Langevin dynamics of a many-body system of interacting particles in $d$ dimensions, in a very general setting suitable to model several out-of-equilibrium situations, such as liquid and glass rheology, active self-propelled particles, and glassy aging dynamics. The pair interaction potential is generic, and can be chosen to model colloids, atomic liquids, and granular materials. In the limit ${dtoinfty}$, we show that the dynamics can be exactly reduced to a single one-dimensional effective stochastic equation, with an effective thermal bath described by kernels that have to be determined self-consistently. We present two complementary derivations, via a dynamical cavity method and via a path-integral approach. From the effective stochastic equation, one can compute dynamical observables such as pressure, shear stress, particle mean-square displacement, and the associated response function. As an application of our results, we derive dynamically the `state-following equations that describe the response of a glass to quasistatic perturbations, thus bypassing the use of replicas. The article is written in a modular way, that allows the reader to skip the details of the derivations and focus on the physical setting and the main results.
It was recently claimed that on d-dimensional small-world networks with a density p of shortcuts, the typical separation s(p) ~ p^{-1/d} between shortcut-ends is a characteristic length for shortest-paths{cond-mat/9904419}. This contradicts an earlier argument suggesting that no finite characteristic length can be defined for bilocal observables on these systems {cont-mat/9903426}. We show analytically, and confirm by numerical simulation, that shortest-path lengths ell(r) behave as ell(r) ~ r for r < r_c, and as ell(r) ~ r_c for r > r_c, where r is the Euclidean separation between two points and r_c(p,L) = p^{-1/d} log(L^dp) is a characteristic length. This shows that the mean separation s between shortcut-ends is not a relevant length-scale for shortest-paths. The true characteristic length r_c(p,L) diverges with system size L no matter the value of p. Therefore no finite characteristic length can be defined for small-world networks in the thermodynamic limit.
The spatially uniaxially anisotropic d=3 Ising spin glass is solved exactly on a hierarchical lattice. Five different ordered phases, namely ferromagnetic, columnar, layered, antiferromagnetic, and spin-glass phases, are found in the global phase diagram. The spin-glass phase is more extensive when randomness is introduced within the planes than when it is introduced in lines along one direction. Phase diagram cross-sections, with no Nishimori symmetry, with Nishimori symmetry lines, or entirely imbedded into Nishimori symmetry, are studied. The boundary between the ferromagnetic and spin-glass phases can be either reentrant or forward, that is either receding from or penetrating into the spin-glass phase, as temperature is lowered. However, this boundary is always reentrant when the multicritical point terminating it is on the Nishimori symmetry line.
In this work we compare and characterize the behavior of Langevin and Dissipative Particle Dynamics (DPD) thermostats in a broad range of non-equilibrium simulations of polymeric systems. Polymer brushes in relative sliding motion, polymeric liquids in Poiseuille and Couette flows, and brush-melt interfaces are used as model systems to analyze the efficiency and limitations of different Langevin and DPD thermostat implementations. Widely used coarse-grained bead-spring models under good and poor solvent conditions are employed to assess the effects of the thermostats. We considered equilibrium, transient, and steady state examples for testing the ability of the thermostats to maintain constant temperature and to reproduce the underlying physical phenomena in non-equilibrium situations. The common practice of switching-off the Langevin thermostat in the flow direction is also critically revisited. The efficiency of different weight functions for the DPD thermostat is quantitatively analyzed as a function of the solvent quality and the non-equilibrium situation.
We consider the spatial correlation function of the two-dimensional Ising spin glass under out-equilibrium conditions. We pay special attention to the scaling limit reached upon approaching zero temperature. The field-theory of a non-interacting field makes a surprisingly good job at describing the spatial shape of the correlation function of the out-equilibrium Edwards-Anderson Ising model in two dimensions.
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