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Typicality paradigm in Quantum Statistical Thermodynamics Barbara Fresch, Giorgio Moro Dipartimento Scienze Chimiche Università di Padova Outline 1) The framework: microcanonical statistics versus the dynamics of an isolated quantum system 2) Equilibrium properties and fluctuations 3) What populations? Statistical Ensemble 4) Typicality 5) Thermodynamics 6) Conclusions: an open issue and further developments 1) The framework Dynamics of a quantum pure state for an isolated system (no interactions and entanglement with the surrounding, otherwise the system’s wavefunction does not exist) H : Hilbert space Ĥ: Hamiltonian operator for the energy Hamiltonian (energy) eigenvalues Ek and eigenstates Ek Hˆ Ek = Ek Ek for k = 1, 2, Schroedinger equation for the evolution of the wavefunction Ψ (t ) ∂ i Ψ (t ) = Hˆ Ψ (t ) with Ψ (t ) | Ψ (t ) = 1 ∂t Density matrix operator: ρˆ (t ) := Ψ (t ) Ψ (t ) Quantum observable a(t ): expectation value of operator Aˆ a(t ) := Ψ (t ) Aˆ Ψ (t ) = Tr { Aˆ ρˆ (t )} Partition of the overall system as subsystem+envinronment: reduced density matrix µˆ (t ) of subsystem µˆ (t ) := Trenvir. { ρˆ (t )} 1) The framework Microcanonical density matrix operator ρˆ µC : Ek ρˆ µC Ek ' = δ k ,k ' ρ kµ,Ck µC for Emin < Ek ≤ Emax µC = 1/ N ρ k ,k otherwise = 0 where ∆E := Emax − Emin is small enough but not too much (Emax ⇒ internal energy U of Thermodynamics) µC Conventional interpretation of ρˆ : average of ρ (t ) amongst a collection of system’s copies 1) The framework In recent years: 1) quantum computer theory 2) decoherence program (Zurek) Statistical Thermodynamics of a single system evolving according to the Schroedinger equation 2) Equilibrium properties and fluctuations Quantum dynamics in a finite dimensional Hilbert space H N defined by an upper energy cut-off Emax H N := span { Ek | Ek ≤ Emax } Wavefunction expansion on the energy eigenstates N Ψ (t ) = ∑ ck (t ) Ek ck (t ) = Ek | Ψ (t ) = e − iEk t / ck (0) k =1 − iα k ( t ) ( ) = e c t P Polar representation of the coefficients: k k 2 Populations: Pk = ck (t ) = ck (0) 2 (constants of motion!) Phases: α k (t ) = α k (0) + Ek t / Pure State Distribution (PSD): homogeneous distribution of the phases for a given set of populations. 2) Equilibrium properties and fluctuations TN ∆ N −1 B 2) Equilibrium properties and fluctuations Randomly perturbed harmonic oscillators: results for the ground state element of the reduced density matrix 0.5204 0.4756 µ1,1 µ1,1 0.4754 0.5202 0.4752 0.52 0 1 2 3 t 8000 4 x 10 0 1 2 4 3 t 4 x 10 8000 5 4 4000 4000 0.4751 5 0.4753 0.4755 µ1,1 0.5199 0.5201 µ1,1 0.5203 Two different (random) choices of the populations (or of Ψ (0) ) 2) Equilibrium properties and fluctuations Equilibrium properties from the asymptotic time average a := limτ →∞ 1 τ ∫ τ 0 a (t )dt = Tr { Aˆ ρˆ } = a ( P) N ρˆ = ∑ Pk Ek Ek k =1 Equilibrium properties depend on the choice of the populations! Note: PSD allows the calculations of fluctuations about the equilibrium value Population dependence also for the (Shannon) entropy S := k B ∑ k =1 Pk ln(1/ Pk ) N and the (internal) energy N U := Ψ (t ) Hˆ Ψ (t ) = ∑ k =1 Pk Ek 3) What populations? No way of imposing populations to a macroscopic system! How to get predictions about a system if the populations are unknown? The populations can be characterized only at the statistical level! Statistical Ensemble for the populations that is Sample Space D + probability density p ( P ) for the populations P = ( P1 , P2 , , PN ) Sample space: simplex in N dimensions (∑ N ) P =1 k =1 k Average of f ( P ) on the Ensemble: 1 1− P1 0 0 f = ∫ dP1 ∫ dP2 ∫ 1− P1− PN −2 0 dPN −1 [ f ( P ) p( P ) ]P No a priori information on p ( P ) ! N =1− P1− PN −1 3) What populations? From the lack of knowledge Random Pure State Ensemble (RPSE): homogeneous distribution of Ψ (0) on the unit sphere in H N z = Ψ (0) y x x, y, z = real or imaginary parts of ck (0) 3) What populations? ( From the (Euclidean) measure in H N: p ( P ) = ( N − 1)!δ 1 − ∑ k =1 Pk N ) Note: RPSE is defined only in a finite dimensional Hilbert space necessity of the high energy cut-off Emax From p ( P ) : statistical sampling of the populations and of the system’s (equilibrium) properties 3) What populations? Example for randomly perturbed harmonic oscillators 800 -4.5 n = number of oscillators 700 -5 8 600 ( ) p µ1,1 -5.5 σµ 1,1 -6 500 σf = -6.5 400 -7 300 5 n=5 0 0.31 6 7 100 0.315 0.32 n 7 8 6 0.325 0.33 0.335 µ1,1 f ) = variance of f ( P) -7.5 200 (f − 2 0.34 0.345 0.35 0.355 0.36 3) What populations? Different realizations of the system lead to different values of the equilibrium properties, in particular of the Entropy and of the Internal Energy What a relation with Thermodynamical properties? Note: standard microcanonical analysis corresponds to a specific choice of the populations µC 1/ N for Emin < Ek ≤ Emax = µC Pk otherwise = 0 4) Typicality Concept of typicality from Information Theory. A is a typical event in a broad meaning if i) Prob { A} = 1 − ε with ε <<1, that is Prob { A} =ε Strong form of typicality if ii) Prob { A} = 1 like the measure/probability of irrational numbers in the interval [ 0,1] In our case: i) Pro b { µ11 − µ11 ≤ 2σ µ11 } 0.99 ii) Lim n→∞ Prob { µ11 − µ11 ≤ K } = 1 ∀ K > 0 Qualitatively speaking, the overwhelming majority of systems has a typical value of the property. 4) Typicality The same behavior is found for the internal energy and the entropy Entropy Distribution for systems of n spins n = 15 -1 10 σ -2 10 n = 14 -3 10 n = 13 -4 10 0 5 n 10 15 n = 11 n = 10 0.645 n = 12 0.650 0.655 0.660 0.665 Entropy per spin S/n For large enough systems, the variability of populations does not influence substantially the observables which, then, can be identified with their Ensemble averages S U µˆ 5) Thermodynamics Is the RPSE description consistent with the equilibrium thermodynamics of macroscopic system? The answer is positive since, for a generic system characterized through its density of states g ( E ) = δ ( E − Ek ) ∑ the following items are derived k a) Typicality in the thermodynamic limit (number of components n → ∞ ) which allows the identification of the entropy ( S ), of the internal energy ( U ) and of the properties of a component trough µ̂ b) Entropy equation of state S = S ( U ), by eliminating the dependence on the energy cut-off Emax S = f ( Emax ) U = g ( Emax ) S = f ( g −1 ( U )) 5) Thermodynamics c) Both S and U are extensive properties therefore the Temperature is an intensive parameter 1 d S := T d U d) S = S ( U ) is a convex increasing function of U , therefore both S and U are increasing functions of the Temperature e) The canonical form for the reduced density matrix of subsystem µˆ ∝ exp {− Hˆ ss / k BT } Hˆ ss : Hamiltonian of the subsystem 6) Conclusions An open issue 1) There are other possible Statistical Ensembles for the populations. In the past the Fixed Expectation Energy Ensemble (FEEE) having the constraint < Ψ | Hˆ | Ψ >= E = constant, has been proposed as the quantum counterpart of the classical microcanonical statistical mechanics. 2) Consistency with Thermodinamics, points a) to e), as the main criterion of choice between Ensembles. As a matter of fact, FEEE does not satisfy conditions d) and e)! 3) Further conditions to be satisfied? Thermalization Switch on the interaction Hamiltonian + 1 spin 9 spins H = H1 + H E + λ H SE Solve the Schrödinger equation ψ S ( 0) = β ψ E ( 0 ) = ∑ k =1 Pk eiα M kmax k ψ (0) = ψ S ( 0 ) ⊗ψ E ( 0 ) ψ E (0) from RPSE 0 -0.05 5 Emax -0.1 -0.15 0 E z M (0) ∝ H E = 0 Mpzz -0.2 -0.25 -0.3 -0.35 -5 -0.4 M z (0) ∝ H S = = Tr ( S Z µ ( 0 ) ) = −0.5 -0.45 -0.5 0 500 1000 t 1500 5 ψ S ( 0 ) = 0.34eiα β + 0.66eiα α 1 2 ψ E (0) from RPSE 0 E z M (0) ∝ H E = −0.66 M zS (0) ∝ H S = 0.16 Thermalization Emax -5 One observes relaxation toward a well defined equilibrium state (i.e. within the statistical uncertainty due to the finiteness of the system) The equilibrium state depends only on the total energy of the system, not on the initial state and equipartition of the energy is observed 0.1 0 -0.1 Mpzz -0.2 -0.3 -0.4 -0.5 0 500 1000 t 1500