86
S. Goto and H. Hino
Finally, to obtain the solution to (4.26) for ℘ θ , one rewrites
℘ θ (s, t) with p θ (s, 0) = 1
as
℘ θ (s, t) = e
γ (s−1) t
℘ θ (s, 0) =
∞
j=0
(γ s t)
j
j !
e
−γ t
=
∞
j=0
(γ t)
j
j !
e
−γ t
s
j
.
Since the term [· · · ] of the equation above should be equal to ℘ θ ( j, t), one has
℘ θ ( j, t) =
(γ t)
j
j !
e
−γ t
,
j = 0, . . . , ∞,
where this distribution function is known as the Poisson distribution with the parameter γ t.
References
1. Amari, S., Nagaoka, H.: Methods of Information Geometry. Oxford University Press, AMS
(2000)
2. Ay, N., et al.: Information Geometry. Springer (2017)
3. Goto, S.: Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics. J. Math. Phys. 56, 073301, 30 pp. (2015)
4. Nakamura, Y.: Gradient systems associated with probability distribution. Jpn. J. Ind. App.
Math. 11, 21–30 (1994)
5. Fujiwara, A., Amari, S.I.: Gradient systems in view of information geometry. Physica D 80,
317–327 (1995)
6. Pistone, G.: Information geometry of the probability simplex: A short course arXiv:1911.01876
7. Arnold, V.I.: Methods of Classical Mechanics. Springer (1976)
8. Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Springer (1987)
9. da Silva, A.C.: Lectures on Symplectic Geometry, 2nd edn. Springer (2008)
10. Hermann, R.: Geometry. Dekker, Systems and Physics (1973)
11. Mrugala, R.: Geometrical formulation of equilibrium phenomenological thermodynamics. Rep.
Math. Phys. 14, 419–427 (1978)
12. Mrugala, R.: On contact and metric structures on thermodynamic spaces. Suken kokyuroku
1142, 167–181 (2000)
13. Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics I: Beltrami fields and the Seifert
conjecture. Nonlinearity 13, 441–458 (2000)
14. Ohsawa, T.: contact geometry of the Pontryagin maximum principle. Automatica 55, 1–5
(2015)
15. Bravetti, A., Cruz, H., Tapias, D.: Contact Hamiltonian mechanics. Ann. Phys. 376, 17–39
(2017)
16. Leon, M., Valcazar, M.L.: Contact Hamiltonian systems. J. Math. Phys. 60, 102902 (2019)
17. Grmela, M.: Contact geometry of mesoscopic thermodynamics and dynamics. Entropy 16,
1652–1686 (2014)
18. van der Schaft, A.J., Maschke, B.: Geometry of thermodynamic process. Entropy, 20, 925 [ 23
pp.], (2018)
19. Goto, S.: Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium thermodynamics. J. Math. Phys. 57, 102702
[40 pp.], (2016)
S. Goto and H. Hino
Finally, to obtain the solution to (4.26) for ℘ θ , one rewrites
℘ θ (s, t) with p θ (s, 0) = 1
as
℘ θ (s, t) = e
γ (s−1) t
℘ θ (s, 0) =
∞
j=0
(γ s t)
j
j !
e
−γ t
=
∞
j=0
(γ t)
j
j !
e
−γ t
s
j
.
Since the term [· · · ] of the equation above should be equal to ℘ θ ( j, t), one has
℘ θ ( j, t) =
(γ t)
j
j !
e
−γ t
,
j = 0, . . . , ∞,
where this distribution function is known as the Poisson distribution with the parameter γ t.
References
1. Amari, S., Nagaoka, H.: Methods of Information Geometry. Oxford University Press, AMS
(2000)
2. Ay, N., et al.: Information Geometry. Springer (2017)
3. Goto, S.: Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics. J. Math. Phys. 56, 073301, 30 pp. (2015)
4. Nakamura, Y.: Gradient systems associated with probability distribution. Jpn. J. Ind. App.
Math. 11, 21–30 (1994)
5. Fujiwara, A., Amari, S.I.: Gradient systems in view of information geometry. Physica D 80,
317–327 (1995)
6. Pistone, G.: Information geometry of the probability simplex: A short course arXiv:1911.01876
7. Arnold, V.I.: Methods of Classical Mechanics. Springer (1976)
8. Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Springer (1987)
9. da Silva, A.C.: Lectures on Symplectic Geometry, 2nd edn. Springer (2008)
10. Hermann, R.: Geometry. Dekker, Systems and Physics (1973)
11. Mrugala, R.: Geometrical formulation of equilibrium phenomenological thermodynamics. Rep.
Math. Phys. 14, 419–427 (1978)
12. Mrugala, R.: On contact and metric structures on thermodynamic spaces. Suken kokyuroku
1142, 167–181 (2000)
13. Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics I: Beltrami fields and the Seifert
conjecture. Nonlinearity 13, 441–458 (2000)
14. Ohsawa, T.: contact geometry of the Pontryagin maximum principle. Automatica 55, 1–5
(2015)
15. Bravetti, A., Cruz, H., Tapias, D.: Contact Hamiltonian mechanics. Ann. Phys. 376, 17–39
(2017)
16. Leon, M., Valcazar, M.L.: Contact Hamiltonian systems. J. Math. Phys. 60, 102902 (2019)
17. Grmela, M.: Contact geometry of mesoscopic thermodynamics and dynamics. Entropy 16,
1652–1686 (2014)
18. van der Schaft, A.J., Maschke, B.: Geometry of thermodynamic process. Entropy, 20, 925 [ 23
pp.], (2018)
19. Goto, S.: Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium thermodynamics. J. Math. Phys. 57, 102702
[40 pp.], (2016)