The general reason for approach to thermal equilibrium of macroscopic quantu

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1 The general reason for approach to thermal equilibrium of macroscopic quantum systems 10 May 2011 Joint work with S. Goldstein, J. L. Lebowitz, C. Mastrodonato, and N. Zanghì.

2 Claim macroscopic quantum system S (say, N > particles) in a bounded volume Λ R 3 isolated, evolves unitarily i ψ t = Hψ H = α E α φ α φ α For most H it is the case that for every initial state ψ 0, the system will spend most of the time (as t ) in thermal equilibrium.

3 What is thermal equilibrium? Two views (in both quantum and classical mechanics) Individualist view A system is in thermal equilibrium if it is in an appropriate pure state (given by a wave function or point in phase space). Ensemblist view A system is in thermal equilibrium if it is in an appropriate statistical state (given by a density matrix or probability measure on phase space). For discussion, see [Goldstein, Lebowitz, RT, Zanghì: Eur. Phys. J. H 35 (2010)]

4 Individualist equilibrium in classical mechanics State: point X = (q 1,..., q N, p 1,..., p N ) in phase space energy shell Γ = {X : E H(X ) E + δe} depending on a choice of macro-variables, partition Γ into macro-states Γ ν corresponding to different (small ranges of) values of the macro-variables, Γν Γeq or rather: Γ = ν Γ ν one cell Γ eq has the overwhelming majority of volume, vol Γ eq vol Γ 1. Def: A system is in equilibrium its phase point lies in the set Γ eq. Γeq

5 Individualist equilibrium in quantum mechanics State: wave fct ψ = ψ(q 1,..., q N ), ψ = 1. Hamiltonian H = α E α φ α φ α { } energy shell H = span φ α : E E α E + δe. dim H macro-states correspond to subspaces H ν, mutually orthogonal, H = ν H ν thermal equilibrium subspace H eq H with dim H eq dim H 1 Def: A system is in thermal equilibrium ψ is close to H eq ψ P eq ψ 1 (where P eq = projection to H eq ) Equilibrium is typical: ψ P eq ψ 1 for most ψ. Proof: E ψ ψ P eq ψ = dim H eq / dim H 1.

6 Ensemblist equilibrium Classical mechanics Def: A system is in thermal equilibrium its state X is random with probability density Quantum mechanics Def: A system is in thermal equilibrium its quantum state is a mixture with density matrix ρ = ρ can = 1 Z e βh (canonical ensemble) or ρ = ρ mc (micro-canonical ensemble). ρ = ρ can = 1 Z e βh (canonical ensemble) or ρ = ρ mc (micro-canonical ensemble). This formulation has the defect that an individual system can t be in equilibrium and the virtue of being precise. It also has the virtue that it admits of a clean simple notion of approach to equilibrium.

7 Approach to thermal equilibrium In the ensemblist view ρ t ρ can or ρ t ρ mc as t Classical mechanics: mixing Quantum mechanics: need open system, coupled to infinite heat bath (otherwise, no convergence by Poincaré recurrence) In the individualist view It is impossible that ψ t P eq ψ t 1 for all t, by Poincaré recurrence. Instead, approach to equilibrium means that ψ t P eq ψ t 1 for most t.

8 Mathematical result (1) Let dim H <, H eq H any subspace, P eq projection to H eq. Theorem 1: Approach to individualist equilibrium Let η, δ > 0, ε = ηδ /2. If and H is non-degenerate (1) φ α P eq φ α > 1 2ε α = 1,..., dim H, (2) then any ψ 0 H with ψ 0 = 1 will spend (1 δ )-most of the time in thermal equilibrium, i.e., 1 { lim inf T T 0 < t < T : ψ t P eq ψ t > 1 η} > 1 δ. M = Lebesgue measure of M [Goldstein, Lebowitz, Mastrodonato, RT, Zanghì: Phys.Rev.E 2010] Condition (2): eigenstate thermalization [Srednicki: Phys.Rev.E 1994]

9 Proof 1 T time average f (t) = lim f (t) dt T T 0 ψ t P eq ψ t =? ψ 0 = dim H α=1 c α φ α, ψ t = ψ t P eq ψ t = α,β = α dim H α=1 > 1 ηδ e ieαt c α φ α e i(eα E β)t }{{} δ αβ c αc β φ α P eq φ β c α 2 φ α P eq φ α }{{} >1 ηδ If error(t) > η for more than δ of the time then error(t) > ηδ. Thus, ψ t P eq ψ t > 1 η for (1 δ )-most of the time.

10 Mathematical result To repeat: Theorem 1 If 2 conditions on H are satisfied, then for every ψ 0, the system spends most of the time in thermal equilibrium.

11 Mathematical result (2) Theorem 2 Let ε, δ > 0, E α pairwise distinct real values, {φ α } a random ONB of H with uniform distribution, H = α E α φ α φ α. If dim H > D 0 (ε, δ) and dim H eq dim H > 1 ε, then condition (2) is (1 δ)-typically satisfied, i.e., { } Prob φ α P eq φ α > 1 2ε α = 1,..., dim H > 1 δ. [Goldstein, Lebowitz, Mastrodonato, RT, Zanghì: Phys.Rev.E 2010] I suggest that this is the general reason for approach to equilibrium. There do exist exceptional Hamiltonians that don t satisfy condition (2) (e.g., non-interacting, Anderson localization).

12 Probability, typicality, universality Talking about probability is just a way of saying what s true of most Hamiltonians; doesn t mean that H is so distributed in nature.

13 What is the meaning of the canonical density matrix ρ can = 1 Z e βh for an individualist? This: Canonical typicality Let system 1 and system 2 consist of N 1 N 2 particles. Let the interaction between systems 1 and 2 be weak, H H 1 I 2 + I 1 H 2. Let H mc H 1 H 2 be an energy shell. For most ψ H mc with ψ = 1, for suitable β = β(e). tr 2 ψ ψ 1 Z e βh1 [Gemmer, Mahler, Michel: book, Springer 2004] [Popescu, Short Winter: Nature Physics 2006] [Goldstein, Lebowitz, RT, Zanghì: Phys.Rev.Lett. (2006)]

14 Entropy Entropy of ensemble Classical: Gibbs entropy S G = k R 6N dq dp ρ(q, p) log ρ(q, p) Quantum: von Neumann entropy S vn = k tr(ρ log ρ) Problematical for an individualist because S = 0 for a pure state. Entropy of individual system Classical: partition Γ = ν Γ ν of phase space Boltzmann entropy S B (X ) = k log vol(γ ν ) if X Γ ν Quantum: orthogonal decomposition H = ν H ν of Hilbert space quantum Boltzmann entropy S qb (ψ) = k log dim H ν if ψ H ν Most ψs are superpositions of states with different values of entropy. It s not obvious what it even means to say that S qb increases with time. Open problem: formulate (and justify) the second law (entropy increase) for S qb.

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30 Thank you for your attention