Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains (2403.11525v2)

Published 18 Mar 2024 in cond-mat.stat-mech and math.PR

Abstract: The large deviations at various levels that are explicit for Markov jump processes satisfying detailed-balance are revisited in terms of the supersymmetric quantum Hamiltonian $H$ that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density ${\hat p}(C) $ of the configurations $C$ seen during a trajectory over the large time-window $[0,T]$ and rewrite the explicit Donsker-Varadhan rate function as the matrix element $I{[2]}[{\hat p}(.) ] = \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat p}} \rangle $ involving the square-root ket $\vert \sqrt{{\hat p}} \rangle $. [The analog formula is also discussed for reversible diffusion processes as a comparison.] We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density ${\hat p}(C) $ and the empirical local activities ${\hat a}(C,C') $ characterizing the density of jumps between two configurations $(C,C')$. Finally, the explicit rate function for the joint probability of the empirical density ${\hat p}(C) $ and of the empirical total activity ${\hat A} $ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements $ \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat p}} \rangle$ and $\langle \sqrt{ {\hat p} } \vert H{off} \vert \sqrt{{\hat p}} \rangle $, where $H{off} $ represents the off-diagonal part of the supersymmetric Hamiltonian $H$. This general formalism is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian $H$ correspond to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (40)
  1. B. Derrida, J. Stat. Mech. P07023 (2007).
  2. R J Harris and G M Schütz, J. Stat. Mech. P07020 (2007).
  3. A. Lazarescu, J. Phys. A: Math. Theor. 48 503001 (2015).
  4. A. Lazarescu, J. Phys. A: Math. Theor. 50 254004 (2017).
  5. A. de La Fortelle, PhD Thesis (2000) ”Contributions to the theory of large deviations and applications” INRIA Rocquencourt.
  6. V. Lecomte, PhD Thesis (2007) ”Thermodynamique des histoires et fluctuations hors d’équilibre” Université Paris 7.
  7. R. Chétrite, PhD Thesis (2008) ”Grandes déviations et relations de fluctuation dans certains modèles de systèmes hors d’équilibre” ENS Lyon.
  8. D. Simon, J. Stat. Mech. (2009) P07017.
  9. R. Chétrite and H. Touchette Ann. Henri Poincare 16, 2005 (2015).
  10. R. Chétrite and H. Touchette, J. Stat. Mech. P12001 (2015).
  11. C. Monthus, J. Stat. Mech. (2019) 023206.
  12. K. Proesmans and B. Derrida, J. Stat. Mech. (2019) 023201.
  13. C. Monthus, J. Stat. Mech. (2021) 033303.
  14. C. Monthus, J. Stat. Mech. (2021) 063301.
  15. C. Monthus, J. Stat. Mech. (2023) 083204
  16. M. Polettini, J. Phys. A: Math. Theor. 48 (2015) 365005.
  17. C. Monthus, J. Stat. Mech. (2021) 033201.
  18. C. Monthus, J. Stat. Mech. (2021) 063211.
  19. C. Monthus, J. Stat. Mech. (2024) 013208.
  20. C. Monthus, J. Stat. Mech. (2022) 013206.
  21. C. Monthus, J. Stat. Mech. (2023) 063206.
  22. C. Monthus, J. Stat. Mech. (2024) 013206.
  23. C. Monthus, 2024 J. Phys. A: Math. Theor. 57 095002.
  24. C. Monthus, J. Stat. Mech. (2024) 013205.
  25. C. Monthus, J. Stat. Mech. (2021) 083212.
  26. C. Monthus, J. Stat. Mech. (2021) 083205
  27. C. Monthus, J. Stat. Mech. (2021) 103202.
  28. C. W. Gardiner, “ Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences” (Springer Series in Synergetics), Berlin (1985).
  29. N.G. Van Kampen, “Stochastic processes in physics and chemistry”, Elsevier Amsterdam (1992).
  30. H. Risken, “The Fokker-Planck equation : methods of solutions and applications”, Springer Verlag Berlin (1989).
  31. C. Monthus and P. Le Doussal, Phys. Rev. E 65 (2002) 66129.
  32. C. Texier and C. Hagendorf, Europhys. Lett. 86 (2009) 37011.
  33. C. Monthus and T. Garel, J. Stat. Mech. P12017 (2009).
  34. C. Monthus, J. Stat. Mech. (2023) 063206
  35. C. L. Henley, 2004 J. Phys.: Condens. Matter 16 S891
  36. J. Kurchan, J. Phys. A: Math. Gen. 31 3719 (1998).
  37. J. Kurchan J. Stat. Mech. (2007) P07005.
  38. R. K. P. Zia and B. Schmittmann J. Stat. Mech. P07012 (2007).
  39. R.L. Jack and P. Sollich, J. Phys. A: Math. Theor. 47 (2014) 015003.
  40. C. Monthus, arXiv:2404.16605.

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com