Typicality and thermalization in isolated quantum systems Hal Tasaki will be revised soon! YKIS 2015 , Aug. 19, 2015 arXiv:1507.06479
about the talk Foundation of equilibrium statistical mechanics based on pure quantum mechanical states in macroscopic isolated quantum systems von Neumann 1929, Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi 2010 Main messages a pure quantum mechanical state can fully represent thermal equilibrium the unitary time evolution in an isolated quantum system can describe thermalization
a pure state which represents thermal equilibrium: an instructive example Choose momenta p 1 , p 2 , . . . , p N randomly according to the Maxwell-Boltzmann distribution at temperature T , and fix them Then, define a pure state by N h i i p ` · r ` Y ϕ ex ( r 1 , . . . , r N ) = exp ~ ` =1 | ϕ ex i Can you experimentally distinguish from the canonical distribution of a dilute gas? Usually, you CAN’T You CAN, IF you know and can | ϕ ex ih ϕ ex | measure the operator The state represents thermal equilibrium! | ϕ ex i
Heuristic pictures about thermal equilibrium
Macroscopic view An isolated macroscopic system always settles to thermal equilibrium state after a sufficiently long time Thermal equilibrium No macroscopic changes, no macroscopic flows Uniquely determined by specifying only few macroscopic variables (e.g., the total energy U , in a system consisting of a single substance when V and N are fixed)
Microscopic view Microscopically there are A LOT OF states with energy U All the micro-states with energy U ( r 1 , r 2 , . . . , r N , p 1 , . . . , p N ) the positions and momenta of all the molecules Standard procedure of statistical mechanics (principle of equal weights) The microcanonical distribution (in which all the micro- states with U appear with the equal probabilities) describes thermal equilibrium Why does this work?? What is the underlying picture?
Typicality argument All the micro-states with energy U FACT: In a macroscopic system, a great majority of microscopic states with energy U look identical from the macroscopic point of view we shall prove this POSTULATE: “thermal equilibrium” = common properties shared by these majority of states thus the microcanonical ensemble works A single microscopic state may fully represent thermal equilibrium!
Thermalization All the micro-states with energy U we shall partially prove this Non-equilibrium states: exceptional thermalization states in the overwhelming majority (thermal equilibrium) Thermalization (= the approach to thermal equilibrium) is quite a robust phenomenon
Some remarks about the basic setting
Basic setting Standard (and realistic) treatment surrounding environment (bigger system) quantum system of interest Our (obviously unrealistic) treatment quantum system of interest perfectly isolated from the outside world
Why isolated systems? Standard (fashionable) answer We can realize isolated quantum systems in ultra cold atoms –7 7 clean system of 10 atoms at 10 K My (old-fashioned) answer This is still a very fundamental study, very very far from practical applications We wish to learn what isolated systems can do (e.g., whether they can thermalize) After that, we may study the effect played by the environment
Settings and main assumptions
The system Isolated quantum system in a large volume V ✦ Particle system with constant ρ = N/V ✦ Quantum spin system Hilbert space H tot Hamiltonian ˆ H Energy eigenvalue and the normalized energy eigenstate ˆ H | ψ j i = E j | ψ j i h ψ j | ψ j i = 1 Suppose that one is interested (only) in n extensive quantities ˆ M 1 , . . . , ˆ M n independent of V
Microcanonical energy shell Fix arbitrary and small , and consider the energy ∆ u u eigeneigenvalues such that u − ∆ u ≤ E j /V ≤ u relabel j so that this corresponds to j = 1 , . . . , D D ∼ e σ 0 V microcanonical average of an observable ˆ O D O i mc := 1 X h ˆ h ψ j | ˆ O | ψ j i D j =1 microcanonical energy shell H sh | ψ j i with the space spanned by j = 1 , . . . , D
Pure state which represents thermal equilibrium M 1 , . . . , ˆ ˆ Extensive quantities M n 1 V h ˆ M i i mc equilibrium value m i := lim V ↑∞ | ϕ i 2 H sh DEFINITION: A normalized pure state , for some , represents thermal equilibrium if V > 0 � � δ i h ϕ | ˆ ⇥ � � ˆ � | ϕ i e − α V ⇤ P M i /V � m i fixed const. for all i = 1 , . . . , n fixed const. (precision) projection | ϕ � ˆ if one measures in such , then M i /V � � � ≤ δ i � (measurement result) − m i with probability ≥ 1 − e − α V
Pure state which represents thermal equilibrium Extensive quantities ˆ M 1 , . . . , ˆ M n 1 V h ˆ M i i mc equilibrium value m i := lim V ↑∞ | ϕ � ˆ if one measures in such , then M i /V � � � ≤ δ i � (measurement result) − m i with probability ≥ 1 − e − α V we almost certainly get the equilibrium value! | ϕ � From we get complete information about the thermal equilibrium represents thermal equilibrium! | ϕ �
Basic assumption guarantees that the system is “healthy” Extensive quantities ˆ M i equilibrium value m i precision δ i statement in statistical mechanics THERMODYNAMIC BOUND (TDB): There is a constant , and one has, for any V γ > 0 n D ⇤E X ˆ | ˆ mc ≤ e − γ V ⇥ P M i /V − m i | ≥ δ i i =1 simply says large fluctuation is exponentially rare in the MC ensemble (large deviation upper bound) expected to be valid in ANY uniform thermodynamic phase, but has been proved in limited situations
Example of TDB ( 1 ) Two identical bodies in thermal contact H = ˆ ˆ H 1 + ˆ H 2 + ˆ H int we focus on the energy difference ˆ ˆ H 1 H 2 M = ˆ ˆ H 1 − ˆ n = 1 H 2 THEOEM: Suppose that the system is not at the triple point. Then one has for any that δ > 0 D ⇤E ˆ | ˆ mc ≤ e − γ V ⇥ P M/V | ≥ δ δ 2 γ ' with 2 k B T 2 c ( T ) proof: elementary method in the large deviation theory
Example of TDB (2) General quantum spin chain translation invariant short-range Hamiltonian and observables X X m ( i ) ˆ ˆ ˆ H = h x M i = ˆ x x x THEOEM: For any and any δ 1 , . . . , δ n > 0 u there exists and one has γ > 0 n D ⇤E X ˆ | ˆ mc ≤ e − γ V ⇥ P M i /V − m i | ≥ δ i i =1 for any V corollary of the general theory of Y . Ogata’s
Typicality of pure states which represent thermal equilibrium
Typicality of thermal equilibrium overwhelming majority of states in the energy shell represent thermal equilibrium (in a H sh certain sense) von Neumann 1929 Bocchieri, Loinger 1959 Llyoid 1988 Sugita 2006 Popescu, Short, Winter 2006 Goldstein, Lebowitz, Tunulkam Zanghi 2006 Reimann 2007 we shall formulate our version
Measure on H sh H sh 3 | ϕ i = P D j =1 | α j | 2 = 1 P D a state with j =1 α j | ψ j i C D can be regarded as a point on the unit sphere of a natural (basis independent) measure on is H sh the uniform measure on the unit sphere corresponding average 1 − P D R � j =1 | α j | 2 � d α 1 · · · d α D δ ( · · · ) ( · · · ) := 1 − P D R � j =1 | α j | 2 � d α 1 · · · d α D δ d α := d (Re α ) d (Im α ) From the symmetry j α k = 1 α ∗ D δ j,k
Average over and mc-average H sh | ϕ i = P D ˆ operator normalized state j =1 α j | ψ j i O quantum mechanical expectation value D X h ϕ | ˆ j α k h ψ j | ˆ O | ϕ i = O | ψ k i α ∗ j,k =1 j α k = 1 α ∗ D δ j,k average over H sh X h ϕ | ˆ j α k h ψ j | ˆ O | ϕ i = O | ψ k i α ∗ average over D energy eigenstates j,k average over D = 1 infinitely many X h ψ j | ˆ O | ψ j i = h ˆ O i mc states in the shell D j =1 Another way of looking at the microcanonical average
Typicality of thermal equilibrium provable for Assume Thermodynamic bound (TDB) some models � � δ i h ϕ | ˆ ⇥ � � ˆ � ⇤ P M i /V � m i | ϕ i = D ⇤ E ˆ ⇥ � � ˆ � ≥ δ i � mc ≤ e − γ V P M i /V − m i = | ϕ i 2 H sh THEOREM: Choose a normalized randomly according to the uniform measure on the unit sphere. Then with probability ≥ 1 − e − ( γ − α ) V � � δ i h ϕ | ˆ ⇥ � � ˆ � | ϕ i e − α V ⇤ P M i /V � m i for each i = 1 , . . . , n | ϕ i 2 H sh Almost all pure states represent thermal equilibrium!!
Typicality of thermal equilibrium | ϕ i 2 H sh Almost all pure states represent thermal equilibrium!! FACT: macroscopically, a great majority of states with the same energy look identical POSTULATE: they correspond to thermal equilibrium the microcanonical energy shell H sh nonequilibrium nonequilibrium nonequilibrium thermal equilibrium
Thermalization or the approach to thermal equilibrium
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