# Preprint: Thermalization of rate-independent processes by entropic regularization

Somewhat belatedly, I’ve just uploaded to the arXiv a preprint of a joint paper with Marisol Koslowski, Florian Theil and Michael Ortiz entitled “Thermalization of rate-independent processes by entropic regularization”. The full paper is slated appear in Discrete and Continuous Dynamical Systems – Series S in early 2013.

The topic of the paper is a cute little extension of some of my PhD work, in which we show that the effect of coupling a rate-independent process (a decent model for plastic evolutions such as dry friction) to a heat bath (injecting a bit of statistically disordered energy) is equivalent to “softening” the dissipation potential by taking its Cramer transform (a smoothing and strict convexification procedure often used in large deviations theory).

To make this a bit more precise, suppose that you have an energetic potential E(t, x), depending smoothly (or at least smoothly enough) upon time t ∈ [0, T] and spatial position / state x ∈ ℝn and a dissipation potential Ψ(x), which is a 1-homogenous and convex function of x ∈ ℝn (e.g. a norm). Given an initial condition x0 ∈ ℝn, the Ψ-gradient descent in E is the solution z: [0, T] → ℝn to the differential inclusion

∂Ψ(dz(t)dt) ∋ −DE(t, z(t))

with z(0) = x0. In discrete time, one can approximate such evolutions using a sequence of variational problems; in fact, this sequence of variational problems is how one shows that the continuous-time problem even has a solution. Given that the system has state xi at discrete time ti, the next state xi+1 at discrete time ti+1 is the one that minimizes the Moreau–Yosida functional

E(ti+1, xi+1) − E(ti, xi) + Ψ((xi+1xi)(ti+1ti)).

The resulting continuous-time evolution is called rate-independent because it has no time-scale of its own: if you were to reparametrize the time interval [0, T] and make the energy E vary more slowly or quickly, then the solution of the reparametrized problem would just be the original solution made slower or faster in the same way. (More formally, the solution operator commutes with monotone reparametrizations of time.)

The idea for modelling the effect of a heat bath on this system is to treat the state as a random variable Xi and have the probability density function for Xi+1 given Xi be the minimizer of a functional in which the expected value of the Moreau–Yosida functional competes with an entropy term. The Moreau–Yosida functional is happy (small) if the system follows the original rate-independent trajectory in a perfectly coherent fashion; the entropy term is happy (small) if the probability distribution of the system spreads out all over ℝn in a highly incoherent fashion. What process will result from the competition of these two terms?

The paper identifies the effective process, which has two remarkable characteristics. First of all, the limit process as time step tends to zero is a deterministic process, even though the discrete-time processes (with strictly positive time step) are all bona fide stochastic processes. Secondly, the limit process is in fact another gradient descent; it solves

∂ΨCT(dz(t)dt) = −DE(t, z(t))

where ΨCT denotes the Cramer transform of Ψ, which is a simultaneous smoothing-out and strict convexification of Ψ (which was originally just convex, and had a sharp corner at 0). The result is that the effect of the heat bath is to produce a new rate-dependent deterministic process that behaves like the old rate-independent one at high rates (speeds), but more like a viscous one at low rates.

The devil is, of course, in the details, which are in the paper. 🙂