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Symmetry classes of classical stochastic processes

Lucas Sá, Pedro Ribeiro, Tomaž Prosen, Denis Bernard

Abstract

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.

Symmetry classes of classical stochastic processes

Abstract

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.

Paper Structure

This paper contains 31 sections, 79 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fifteenfold symmetry classification
  3. Analytical families of Markov matrices
  4. Class AI, $S^2=R_+R_+^{-T}=R_-R_-^{-T}=0$
  5. Class AI$_+$, $S^2=+1$, $R_+R_+^{-T}=R_-R_-^{-T}=0$
  6. Class BDI$^\dagger$, $R_+R_+^{-T}=+1$, $R_-R_-^{-T}=S^2=0$
  7. Class CI$_{+-}$, $R_+R_+^{-T}=S^2=+1$, $R_-R_-^{-T}=-1$
  8. Class BDI$_{++}$, $R_+R_+^{-T}=R_-R_-^{-T}=S^2=+1$
  9. Absence of analytic examples for the other classes
  10. Systematic numerical solutions of the symmetry classes
  11. Parametrization of the symmetry operators
  12. Solution of the symmetry constraints
  13. Monte Carlo sampling
  14. Results
  15. Examples and physical consequences
  16. ...and 16 more sections

Figures (2)

  • Figure 1: Results of the stochastic optimization algorithm for all fourteen classes with symmetry constraints, for $N=8$ and $n=2$. Each panel shows the value of the cost function $f$ as a function of the number of acceptances starting from 10 different initial conditions $\{W^{(i)},\bm{s}^{(i)}\}$, with $\delta=1$, $m_\delta=500$, and up to $2\times10^4$ iterations.
  • Figure 2: Spectrum of $L$ in the complex plane for a random Markov matrix from class AI$_+$ (left), sampled according to Eq. (\ref{['eq:parametrization_AI+']}) and which displays dihedral symmetry, and class BDI$^\dagger$ (right), sampled from Eq. (\ref{['eq:parametrization_BDIdg']}) and which does not show dihedral symmetry. In both cases, we set $N=100$ and the entries of the blocks $A$, $B$, $C$ are the absolute values of normal random variables with zero mean and variance $2/N$.