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Stochastic trace formulas

C. P. Dettmann, Gergely Palla, Niels Søndergaard, Gábor Vattay

TL;DR

This work develops a polynomial-basis diagonalization framework to analyze the spectrum of stochastic evolution operators for chaotic systems with additive noise. It constructs a stochastic analogue of the Gutzwiller spectral determinant by performing a weak-noise expansion around classical periodic orbits and organizing trace contributions via prime cycles, enabling higher-order ħ-like corrections to be computed beyond previous capabilities. The authors implement a detailed matrix representation of the Perron-Frobenius operator, derive saddle-point expansions, and test the approach on a 1D expanding map with Gaussian noise, obtaining accurate σ^6–σ^10 corrections and illustrating super-exponential convergence with cycle length. The results provide a practical, generalizable method for evaluating long-time averages and spectra in noisy chaotic systems, with potential applications to more complex flows and higher-dimensional dynamics.

Abstract

The spectrum of the evolution Operator associated with a nonlinear stochastic flow with additive noise is evaluated by diagonalization in a polynomial basis. The method works for arbitrary noise strength. In the weak noise limit we formulate a new perturbative expansion for the spectrum of the stochastic evolution Operator in terms of expansions around the classical periodic orbits. The diagonalization of such operators is easier to implement than the standard Feynman diagram perturbation theory. The result is a stochastic analog of the Gutzwiller semiclassical spectral determinant with the ``$\hbar$'' corrections computed to at least two orders more than what has so far been attainable in stochastic and quantum-mechanical applications, supplemented by the estimate for the late terms in the asymptotic saddlepoint expansions.

Stochastic trace formulas

TL;DR

This work develops a polynomial-basis diagonalization framework to analyze the spectrum of stochastic evolution operators for chaotic systems with additive noise. It constructs a stochastic analogue of the Gutzwiller spectral determinant by performing a weak-noise expansion around classical periodic orbits and organizing trace contributions via prime cycles, enabling higher-order ħ-like corrections to be computed beyond previous capabilities. The authors implement a detailed matrix representation of the Perron-Frobenius operator, derive saddle-point expansions, and test the approach on a 1D expanding map with Gaussian noise, obtaining accurate σ^6–σ^10 corrections and illustrating super-exponential convergence with cycle length. The results provide a practical, generalizable method for evaluating long-time averages and spectra in noisy chaotic systems, with potential applications to more complex flows and higher-dimensional dynamics.

Abstract

The spectrum of the evolution Operator associated with a nonlinear stochastic flow with additive noise is evaluated by diagonalization in a polynomial basis. The method works for arbitrary noise strength. In the weak noise limit we formulate a new perturbative expansion for the spectrum of the stochastic evolution Operator in terms of expansions around the classical periodic orbits. The diagonalization of such operators is easier to implement than the standard Feynman diagram perturbation theory. The result is a stochastic analog of the Gutzwiller semiclassical spectral determinant with the ``'' corrections computed to at least two orders more than what has so far been attainable in stochastic and quantum-mechanical applications, supplemented by the estimate for the late terms in the asymptotic saddlepoint expansions.

Paper Structure

This paper contains 14 sections, 62 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Matrix representation of Perron-Frobenius oper-ator
  3. Expanding maps with a single fixed point
  4. Evolution oper-ator composition for piecewise-analytic expanding maps
  5. Saddle point expansions in terms of prime cycles
  6. Evolution oper-ator s in a matrix representation
  7. Multinomials
  8. Noiseless case
  9. Numerical tests
  10. From traces to $\sigma^{2m}$ corrections
  11. Testing the results
  12. Summary and outlook
  13. Acknowledgements
  14. Algebra

Figures (4)

  • Figure 1: The generalized cumulants $Q_{n j}$ as a function of cycle length $n$ for $j=0,2,4,6,8$. $Q_{n 0}$ is the cumulant of the noiseless case. Superexponential convergence can be observed until cycle length $n=6$, then numerical errors take over.
  • Figure 2: The numerical eigenvalue and the known terms:$\nu(n)=\sum_{k=0}^{n/2} \nu_{2k}\sigma^{2k}$
  • Figure 3: The difference between the numerical eigenvalue and the known terms.
  • Figure 4: The result of numerical integration and the eigenvalue fit together well in the large $\sigma$ region.