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On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures

Diego Alberici, Nicolas Macris, Emanuele Mingione

Abstract

We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula.

On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures

Abstract

We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula.
Paper Structure (22 sections, 43 theorems, 333 equations)

This paper contains 22 sections, 43 theorems, 333 equations.

Table of Contents

  1. Definitions and main result under very simple hypothesis
  2. Generalisation of the main result and logarithmic Sobolev inequalities
  3. Examples
  4. Quadratic potential in dimensions $d_1=d_2=1$: explicit computations
  5. Spin glass: dynamical approach to Guerra's scheme
  6. One-body system ($\Delta=0$)
  7. Interacting system ($\Delta>0$)
  8. High-dimensional inference (rank-one matrix estimation)
  9. Proof of Theorem \ref{['th:main']} and further results
  10. Existence, uniqueness and regularity of the solution
  11. Integrability of $D^\nu V$, $D^\nu F$, $\log\rho$, $\nabla\log\rho$
  12. Fokker-Planck equation for the marginal $\mathop{\mathrm{\rho_{\textit{t},\lambda}^\textup{\tiny(2)}}}\nolimits$
  13. Joint and marginal entropy dissipation
  14. Convergence of the conditional measure $\mathop{\mathrm{\rho_{\textit{t},\lambda}^\textup{\tiny(1)}}}\nolimits$ to $\mathop{\mathrm{\rho_{\!_\textit{\scriptsize*}}^\textup{\tiny(1)}}}\nolimits\,$: proof of Theorem \ref{['th:main']}, first part
  15. Convergence of the marginal measure $\mathop{\mathrm{\rho_{\textit{t},\lambda}^\textup{\tiny(2)}}}\nolimits$ to $\mathop{\mathrm{\rho_{\!_\textit{\scriptsize*}}^\textup{\tiny(2)}}}\nolimits\,$: proof of Theorem \ref{['th:main']}, second part
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\beta_1,\beta_2>0$. Let $V=V_\textup{c}+V_\textup{b}$, where $V_\textup{c}$ is a strongly convex polynomial and $V_\textup{b}$ is smooth, bounded, with bounded derivatives of any order. Let $\mathop{\mathrm{\rho_\textup{I}}}\nolimits$ be a bounded probability density having finite moments of an

Theorems & Definitions (105)

  • Theorem 1.1: Main result under simple hypothesis
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Main result under general hypothesis
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6: tensorisation Gross
  • ...and 95 more