Modern ergodic theory; from a physics hypothesis to a mathematical theory Do˘ganC¸¨omez Department of Physics and Astrophysics, University of North Dakota October 30, 2015 establishing a connection between the ensembles typically studied in statistical mechanics and the properties of single systems. More specifically, in solving problems of demonstrating the equality of infinite time averages and phase averages. Preamble Mathematical study of ergodicity Preamble Origins of Ergodic theory goes back to statistical mechanics; Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory More specifically, in solving problems of demonstrating the equality of infinite time averages and phase averages. Preamble Mathematical study of ergodicity Preamble Origins of Ergodic theory goes back to statistical mechanics; establishing a connection between the ensembles typically studied in statistical mechanics and the properties of single systems. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Preamble Mathematical study of ergodicity Preamble Origins of Ergodic theory goes back to statistical mechanics; establishing a connection between the ensembles typically studied in statistical mechanics and the properties of single systems. More specifically, in solving problems of demonstrating the equality of infinite time averages and phase averages. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory The state of a single particle moving in this space can be described by the trajectory of a point x = (p; q); where N p; q 2 R are position and momenta of all N particles in the system. If the energy of the system is E, then x must lie on the energy surface H(x) = E; where H is the Hamiltonian dq @H dp @H i = ; i = − ; 1 ≤ i ≤ N: dt @pi dt @qi Preamble Mathematical study of ergodicity Consider a physical system of N particles confined in a compact phase space X : Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory If the energy of the system is E, then x must lie on the energy surface H(x) = E; where H is the Hamiltonian dq @H dp @H i = ; i = − ; 1 ≤ i ≤ N: dt @pi dt @qi Preamble Mathematical study of ergodicity Consider a physical system of N particles confined in a compact phase space X : The state of a single particle moving in this space can be described by the trajectory of a point x = (p; q); where N p; q 2 R are position and momenta of all N particles in the system. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory where H is the Hamiltonian dq @H dp @H i = ; i = − ; 1 ≤ i ≤ N: dt @pi dt @qi Preamble Mathematical study of ergodicity Consider a physical system of N particles confined in a compact phase space X : The state of a single particle moving in this space can be described by the trajectory of a point x = (p; q); where N p; q 2 R are position and momenta of all N particles in the system. If the energy of the system is E, then x must lie on the energy surface H(x) = E; Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Preamble Mathematical study of ergodicity Consider a physical system of N particles confined in a compact phase space X : The state of a single particle moving in this space can be described by the trajectory of a point x = (p; q); where N p; q 2 R are position and momenta of all N particles in the system. If the energy of the system is E, then x must lie on the energy surface H(x) = E; where H is the Hamiltonian dq @H dp @H i = ; i = − ; 1 ≤ i ≤ N: dt @pi dt @qi Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Hence, we have a one-parameter continuous flow of 2N transformations τ = fTt gt2R on the phase space X ⊂ R that describes the evolution of the system. 2N So, the orbit of a particle x = (p; q) is Ox = fTt (x)g ⊂ R : By Liouville's Theorem, τ preserves the normalized Lebesgue measure on X : Preamble Mathematical study of ergodicity Given an initial state x; such a system always has a unique solution, which determines the state Tt (p; q) = (p(t); q(t)) at any time t 2 R: Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory 2N So, the orbit of a particle x = (p; q) is Ox = fTt (x)g ⊂ R : By Liouville's Theorem, τ preserves the normalized Lebesgue measure on X : Preamble Mathematical study of ergodicity Given an initial state x; such a system always has a unique solution, which determines the state Tt (p; q) = (p(t); q(t)) at any time t 2 R: Hence, we have a one-parameter continuous flow of 2N transformations τ = fTt gt2R on the phase space X ⊂ R that describes the evolution of the system. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory By Liouville's Theorem, τ preserves the normalized Lebesgue measure on X : Preamble Mathematical study of ergodicity Given an initial state x; such a system always has a unique solution, which determines the state Tt (p; q) = (p(t); q(t)) at any time t 2 R: Hence, we have a one-parameter continuous flow of 2N transformations τ = fTt gt2R on the phase space X ⊂ R that describes the evolution of the system. 2N So, the orbit of a particle x = (p; q) is Ox = fTt (x)g ⊂ R : Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Preamble Mathematical study of ergodicity Given an initial state x; such a system always has a unique solution, which determines the state Tt (p; q) = (p(t); q(t)) at any time t 2 R: Hence, we have a one-parameter continuous flow of 2N transformations τ = fTt gt2R on the phase space X ⊂ R that describes the evolution of the system. 2N So, the orbit of a particle x = (p; q) is Ox = fTt (x)g ⊂ R : By Liouville's Theorem, τ preserves the normalized Lebesgue measure on X : Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory namely, (X ; B; µ); where B is the Borel σ-algebra of subsets of X and µ is the normalized Lebesgue measure. (X ; B; µ, τ) is a measurable as well as topological dynamical system. If f : X ! R denotes a function of a physical quantity, measured during an experiment, for any t ≥ 0; f (Tt x) is the value it takes at the instant t; provided that the system is at x when t = 0: Preamble Mathematical study of ergodicity Structures on X : 2N X ⊂ R is a compact manifold inheriting its topological 2N structure from R : 2N X has a measurable structure inherited from R ; Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory If f : X ! R denotes a function of a physical quantity, measured during an experiment, for any t ≥ 0; f (Tt x) is the value it takes at the instant t; provided that the system is at x when t = 0: Preamble Mathematical study of ergodicity Structures on X : 2N X ⊂ R is a compact manifold inheriting its topological 2N structure from R : 2N X has a measurable structure inherited from R ; namely, (X ; B; µ); where B is the Borel σ-algebra of subsets of X and µ is the normalized Lebesgue measure. (X ; B; µ, τ) is a measurable as well as topological dynamical system. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Preamble Mathematical study of ergodicity Structures on X : 2N X ⊂ R is a compact manifold inheriting its topological 2N structure from R : 2N X has a measurable structure inherited from R ; namely, (X ; B; µ); where B is the Borel σ-algebra of subsets of X and µ is the normalized Lebesgue measure. (X ; B; µ, τ) is a measurable as well as topological dynamical system. If f : X ! R denotes a function of a physical quantity, measured during an experiment, for any t ≥ 0; f (Tt x) is the value it takes at the instant t; provided that the system is at x when t = 0: Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Hence, the result of a measurement is actually the time average of f ; i.e., 1 Z t f (Tt x)dt: t 0 Since macroscopic interval of time for the measurements is extremely large from the microscopic point of view, one may actually consider the limit of the time averages: 1 Z t lim f (Tt x)dt: t!1 t 0 Preamble Mathematical study of ergodicity Boltzmann: The measurements of the precise values of f (Tt x) is not possible since it requires knowing the detailed positions and momenta of all N particles. Do˘ganC¸¨omez Modern ergodic theory; from a physics hypothesis to a mathematical theory Since macroscopic interval of time for the measurements is extremely large from the microscopic point of view, one may actually consider the limit of the time averages: 1 Z t lim f (Tt x)dt: t!1 t 0 Preamble Mathematical study of ergodicity Boltzmann: The measurements of the precise values of f (Tt x) is not possible since it requires knowing the detailed positions and momenta of all N particles.