4 Contact Hamiltonian Systems for Probability Distribution Functions …
87
20. Bravetti, A., Lopez-Monsalvo, C.S.: Para-Sasakian geometry in thermodynamic fluctuation
theory. J. Phys. A: Math. Theor. 48, 125206 [21 pp.], (2015)
21. Bravetti, A., Lopez-Monsalve, C.S., Nettel, F.: Contact symmetries and Hamiltonian thermodynamics. Ann. Phys. 361, 377–400 (2015)
22. van Kampen: Stochastic Processes in Physics and Chemistry. North Holland, (1981)
23. Landau, D., Binder, K.: A guide to Monte-Carlo Simulations in Statistical Physics. Cambridge
Univ Press (2005)
24. Gelman, A. et al.: Beysian Data Analysis 3rd Ed. Chapman and Hal/CRC (2013)
25. Richey, M.: The evolution of Markov Chain Monte Carlo methods. Am. Math. Month. 117,
383–413 (2010)
26. Goto , S., Hino, H.: Diffusion equations from master equations – A discrete geometric approach.
J. Math. Phys. 61, 113301 [27 pp.], (2020). https://aip.scitation.org/doi/10.1063/5.0003656
27. Goto , S., Hino, H.: Information and contact geometric description of expectation variables
exactly derived from master equations. Physica Scripta 95, 015207 [14 pp.], (2020)
28. Goto, S., Hino, H.: Expectation variables on a para-contact metric manifold exactly derived
from master equations. Geom. Sci. Inf. 239–247, (2019)
29. Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Geom. 36, 37–60
(2009)
30. Mrugala, R.: Statistical approach to the geometric structure of thermodynamics. Phys. Rev. A
41, 3156–3160 (1990)
31. Matsuzoe, H., Henmi, M.: Hessian structures on deformed exponential families, GSI 2013.
LNCS 8085, 275–282 (2013)
32. Henmi, M., Matsuzoe, H.: Statistical manifolds admitting torsion and partially flat spaces.
Springer, Geometric Structures of Information. Signal and Communication Technology (2019)
33. Harper, M.: Information geometry and evolutionary game theory. arXiv:0911.1383
34. Ezra, G.S.: Geometric approach to response theory in non-Hamiltonian systems. J. Math. Chem.
32, 339–360 (2002)
35. Ezra, G.S.: On the statistical mechanics of non-Hamiltonian systems: the generalized Liouville
equation, entropy, and time-dependent metrics. J. Math. Chem. 35, 29–53 (2004)
36. Sergi, A., Giaquinta, P.V.: On the geometry and entropy of non-Hamiltonian phase space. J.
Stat. Mech. 2007, PO2013, (2007)
37. Ohara, A., Wada, T.: Information geometry of q-Gaussian densities and behaviors of solutions
to related diffusion equations. J. Phys. A 43, 035002 [18 pp.], (2010)
38. Bravetti, A., Tapias, D.: Liouville’s theorem and the canonical measure for nonconservative
system for contact geometry. J. Phys. A 48, 245001 [11 pp.], (2015)
39. Goto, S., Umeno, K.: Maps on statistical manifolds exactly reduced from the Peron-Frobenius
equations for solvable chaotic maps. J. Math. Phys. 59, 032701 [13 pp.], (2018)
40. Shahshahani, S.: A new mathematical framework for the study of linkage and selection, Memories of AMS, (1979)
41. Suzuki, M.: Statistical mechanics of non-equilibrium systems II Prog. Theo. Phys. 55, 383–399
(1976)
42. Rajeev, S.G.: A Hamilton-Jacobi formalism for thermodynamics. Ann. Phys. 323, 2265–2285
(2008)
43. Balian, R., Valentin, P.: Hamiltonian structure of thermodynamics with guage. Eur. Phys. J. B
21, 269–282 (2001)
44. Gay-Balmaz, F., Yoshimura, H.: Dirac structures in nonequilibrium thermodynamics. J. Math.
Phys. 59, 012701 [29 pp.], (2018)
87
20. Bravetti, A., Lopez-Monsalvo, C.S.: Para-Sasakian geometry in thermodynamic fluctuation
theory. J. Phys. A: Math. Theor. 48, 125206 [21 pp.], (2015)
21. Bravetti, A., Lopez-Monsalve, C.S., Nettel, F.: Contact symmetries and Hamiltonian thermodynamics. Ann. Phys. 361, 377–400 (2015)
22. van Kampen: Stochastic Processes in Physics and Chemistry. North Holland, (1981)
23. Landau, D., Binder, K.: A guide to Monte-Carlo Simulations in Statistical Physics. Cambridge
Univ Press (2005)
24. Gelman, A. et al.: Beysian Data Analysis 3rd Ed. Chapman and Hal/CRC (2013)
25. Richey, M.: The evolution of Markov Chain Monte Carlo methods. Am. Math. Month. 117,
383–413 (2010)
26. Goto , S., Hino, H.: Diffusion equations from master equations – A discrete geometric approach.
J. Math. Phys. 61, 113301 [27 pp.], (2020). https://aip.scitation.org/doi/10.1063/5.0003656
27. Goto , S., Hino, H.: Information and contact geometric description of expectation variables
exactly derived from master equations. Physica Scripta 95, 015207 [14 pp.], (2020)
28. Goto, S., Hino, H.: Expectation variables on a para-contact metric manifold exactly derived
from master equations. Geom. Sci. Inf. 239–247, (2019)
29. Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Geom. 36, 37–60
(2009)
30. Mrugala, R.: Statistical approach to the geometric structure of thermodynamics. Phys. Rev. A
41, 3156–3160 (1990)
31. Matsuzoe, H., Henmi, M.: Hessian structures on deformed exponential families, GSI 2013.
LNCS 8085, 275–282 (2013)
32. Henmi, M., Matsuzoe, H.: Statistical manifolds admitting torsion and partially flat spaces.
Springer, Geometric Structures of Information. Signal and Communication Technology (2019)
33. Harper, M.: Information geometry and evolutionary game theory. arXiv:0911.1383
34. Ezra, G.S.: Geometric approach to response theory in non-Hamiltonian systems. J. Math. Chem.
32, 339–360 (2002)
35. Ezra, G.S.: On the statistical mechanics of non-Hamiltonian systems: the generalized Liouville
equation, entropy, and time-dependent metrics. J. Math. Chem. 35, 29–53 (2004)
36. Sergi, A., Giaquinta, P.V.: On the geometry and entropy of non-Hamiltonian phase space. J.
Stat. Mech. 2007, PO2013, (2007)
37. Ohara, A., Wada, T.: Information geometry of q-Gaussian densities and behaviors of solutions
to related diffusion equations. J. Phys. A 43, 035002 [18 pp.], (2010)
38. Bravetti, A., Tapias, D.: Liouville’s theorem and the canonical measure for nonconservative
system for contact geometry. J. Phys. A 48, 245001 [11 pp.], (2015)
39. Goto, S., Umeno, K.: Maps on statistical manifolds exactly reduced from the Peron-Frobenius
equations for solvable chaotic maps. J. Math. Phys. 59, 032701 [13 pp.], (2018)
40. Shahshahani, S.: A new mathematical framework for the study of linkage and selection, Memories of AMS, (1979)
41. Suzuki, M.: Statistical mechanics of non-equilibrium systems II Prog. Theo. Phys. 55, 383–399
(1976)
42. Rajeev, S.G.: A Hamilton-Jacobi formalism for thermodynamics. Ann. Phys. 323, 2265–2285
(2008)
43. Balian, R., Valentin, P.: Hamiltonian structure of thermodynamics with guage. Eur. Phys. J. B
21, 269–282 (2001)
44. Gay-Balmaz, F., Yoshimura, H.: Dirac structures in nonequilibrium thermodynamics. J. Math.
Phys. 59, 012701 [29 pp.], (2018)