Abstract
We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]ℕ → [0, 1]ℕ, where [0, 1] is the unit closed interval, and the potential A: [0, 1]ℕ → ℝ considered depends on the two first coordinates of [0, 1]ℕ. We are interested in finding stationary Markov probabilities µ∞ on [0, 1]ℕ that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]ℕ. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ β which weakly converges to µ∞. The probabilities µ β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]ℕ, we can consider its projection ν on [0, 1]2. We look at the problem in both ways. If µ∞ is the maximizing probability on [0, 1]ℕ, we also have that its projection ν ∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C 2 and the twist condition, that is,
we show the graph property of the maximizing probability ν on [0, 1]2. Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.
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References
S. Adams. Mathematical Statistical Mechanics. Max-Plank-Institut fur Math., (2006).
K. Athreya and S. Lahiri. Measure Theory and Probability Theory. Springer Verlag, (2006).
V. Bangert. Mather sets for twist maps and geodesics on tori. Dynamics Reported, 1 (1998), 1–56.
I. Brevik, J-M. Borven and S. Ng. Viscous Brane Cohomology with a Brane-Bulk energy interchange term. General Relativity and Gravitation, 38(5) (2006), 907–915(9).
A. Baraviera, A.O. Lopes and Ph. Thieullen. A Large Deviation Principle for equilibrium states of Hölder potentials: the zero temperature case. Stoch. and Dyn., 6 (2006), 77–96.
P. Bhattacharya and M. Majumdar. Random Dynamical Systems. Cambridge Univ. Press, (2007).
W. Chou and R. Griffiths. Ground states of one-dimensional systems using effetive potentials. Physical Review B, 34(9) (1986), 6219–6234.
M. Cveti, S. Nojiri and S.D. Odintsov. Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnetgravity. Nuclear Physics B, 628,Issues 1–2, (2002), 295–330.
P. Bernard and G. Contreras. A Generic Property of Families of Lagrangian Systems. Annals of Math., 167(3) (2008), 1099–1108.
G. Contreras and R. Iturriaga. Global minimizers of autonomous Lagrangians. 22° Colóquio Brasileiro de Matemática, IMPA, (1999).
G. Contreras, A.O. Lopes and Ph. Thieullen. Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory and Dynamical Systems, 21 (2001), 1379–1409.
J.P. Conze and Y. Guivarc’h. Croissance des sommes ergodiques et principe variationnel. Manuscript circa (1993).
P. Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobiequations, and optimal control. Progress in Nonlinear Differential Equations and their Applications 58. Birkhäuser Boston Inc., Boston, MA, (2004).
K. Deimling. Nonlinear Functional Analysis. Springer Verlag, (1985).
C. Dellacherie. Probabilities and potential. North-Holland, (1978).
A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Springer Verlag, (1998).
L.C. Evans. Weak Convergence Methods for Nonlinear Partial Differential Equations. Published for the Conference Board of the Mathematical Sciences, Washington, DC, (1990).
A. Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. Comptes Rendus de l’Académie des Sciences, Série I, Mathématique, 324 (1997), 1043–1046.
A. Fathi and A. Siconolfi. Existence of C 1 critical subsolutions of the Hamilton-Jacobi equations. Inv. Math., 155 (2004), 363–388.
E. Garibaldi and A.O. Lopes. On Aubry-Mather theory for symbolic dynamics. Ergodic Theory and Dynamical Systems, 28,Issue 3 (2008), 791–815.
E. Garibaldi, A.O. Lopes and Ph. Thieullen. On separating sub-actions. Preprint (2006). To appear.
D. Gomes, A.O. Lopes and J. Mohr. The Mather measure and a Large Deviation Principle for the Entropy Penalized Method. Preprint (2007). To appear.
C. Gole. Sympletic twist maps. World Sci. Pub Co Inc., (1998).
D.A. Gomes. Viscosity Solution methods and discrete Aubry-Mather problem. Discrete Contin. Dyn. Syst., 13(1) (2005), 103–116.
D.A. Gomes. Calculus of Variations. IST — Lisboa, (2006).
D.A. Gomes and E. Valdinoci. Entropy Penalization Methods for Hamilton-Jacobi Equations. Adv. Math., 215(1) (2007), 94–152.
E. Hopf. An inequality for Positive Linear Integral Operators. Journal of Mathematics and Mechanics, 12(5) (1963), 683–692.
O. Jenkinson. Ergodic optimization. Discrete and Continuous Dynamical Systems, Series A, 15 (2006), 197–224.
G. Jumarie. Relative Information. Springer Verlag, (1990).
S. Karlin. Total Positivity. Standford Univ. Press, (1968).
E. Lubkin. Negative entropy, energy, and heat capacity in connection with surface tension: Artifact of a model or real?. Inter. Journal of Theoretical Physics, 26(5) (1987), 455–481.
R. Mañé. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 9 (1996), 273–310.
J. Mather. Action minimizing invariant measures for positive definite Lagrangian Systems. Math. Z., 207(2) (1991), 169–207.
I.D. Morris. A sufficient condition for the subordination principle in ergodic optimization. Bull. Lond. Math. Soc., 39(2) (2007), 214–220.
I. Mitra. Introduction to dynamic optimization theory. Optimization and Chaos. Editors M. Majumdar, T. Mitra and K. Nishimura. Springer Verlag, (2000), 31–108.
R.K. Niven. Cost of s-fold Decisions in Exact Maxwell-Boltzmann, Bose-Einsteinand Fermi-DiracStatistics. Physica A, 365,Issue 1 (2006), 142–149.
A. Ostrowski. On positive matrices. Math. Annalen, 150 (1963), 276–284.
M. Pettini. Geometry and topology in Hamiltonian dynamics and statistical mechanics. Springer Verlag, (2007).
W. Parry and M. Pollicott. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, 187–188 (1990).
S. Rachev and L. Ruschendorf. Mass transportation problems, Vol. I and II. Springer Verlag, (1998).
R.T. Rockafellar. Extention of Fenchel’s duality theorem for convex functions. Duke Math. J., 33 (1966), 81–89.
S. Risau-Gusman A.C. Ribeiro-Teixeira and D.A. Stariolo. Topology and Phase Transitions: The Case of the Short Range Spherical Model. Journ. of Statist. Physics, 124(5) (2006), 1231–1253.
H.H. Schaefer. Banach Lattices and Positive Operators. Springer Verlag, (1974).
F. Spitzer. A Variational characterization of finite Markov chains. The Annals of Mathematical Statistics, 43(1) (1972), 303–307.
M. Takahashi. Thermodynamics of one-dimensional solvable models. Cambridge Press, (2005).
C. Thompson. Infinite-Spin Ising Model in one dimension. Journal of Mathematical Physics, 9(2) (1968), 241–245.
A. van Enter, S. Romano and V. Zagrebnov. First-order transitions for some generalized XY models. J. Phys. A., 39(26) (2006), 439–445.
W.F. Wrezinski and E. Abdalla. A precise formulation of the third law of thermodynamics with applications to statistical physics and black holes. Preprint USP (2007).
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Lopes, A.O., Mohr, J., Souza, R.R. et al. Negative entropy, zero temperature and Markov chains on the interval. Bull Braz Math Soc, New Series 40, 1–52 (2009). https://doi.org/10.1007/s00574-009-0001-4
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DOI: https://doi.org/10.1007/s00574-009-0001-4
Keywords
- negative entropy
- Markov chain on [0, 1]
- zero temperature
- penalized entropy
- maximizing probability
- graph property