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) quilibrium properties and fluctuations 3) What populations? Statistical nsemble 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 k and eigenstates k Hˆ k = k k for k = 1, 2, Schroedinger equation for the evolution of the wavefunction Ψ( t) iħ Ψ ( t) = Hˆ Ψ ( t) with Ψ ( t) Ψ ( t) = 1 t Density matrix operator: ˆ ρ( t) : = Ψ( t) Ψ( t) Quantum observable a( t): expectation value of operator Aˆ a( t) : = Ψ( t) Aˆ Ψ ( t) = Tr Aˆ ˆ ρ( t) envir. { ˆ t } { } Partition of the overall system as subsystem+envinronment: reduced density matrix ˆ µ ( t) of subsystem ˆ( µ t) : = Tr ρ( )
1) The framework ˆ µ C Microcanonical density matrix operator : ρ µ C µ C k k ' k, k ' k, k µ C µ C 1/ for min k, k max ˆ ρ = δ ρ = N < k max ρ = otherwise where : = max min is small enough but not too much ( internal energy U of Thermodynamics) Conventional interpretation of ˆ ρ µ C : 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) quilibrium properties and fluctuations Quantum dynamics in a finite dimensional Hilbert space an upper energy cut-off H max { } : = span N k k max H N defined by Wavefunction expansion on the energy eigenstates N ikt/ ħ ( t) ck ( t) k ck ( t) k ( t) e ck () k= 1 i k ( t ) ck ( t) = Pk e α Ψ = = Ψ = Polar representation of the coefficients: 2 2 k k k Populations: P = c ( t) = c () (constants of motion!) Phases: α ( t) = α () + t / ħ k k k Pure State Distribution (PSD): homogeneous distribution of the phases for a given set of populations.
2) quilibrium properties and fluctuations T N B N 1
2) quilibrium properties and fluctuations Randomly perturbed harmonic oscillators: results for the ground state element of the reduced density matrix.4756.524 µ 1,1.4754 µ 1,1.522.4752.52 8 1 2 3 4 5 t x 1 4 8 1 2 3 4 5 t x 1 4 4 4.4751.4753.4755.5199.521.523 µ µ 1,1 1,1 Two different (random) choices of the populations (or of Ψ() )
2) quilibrium properties and fluctuations quilibrium properties from the asymptotic time average 1 τ a : = lim ( ) Tr { ˆ ˆ} τ a t dt Aρ a( P) τ = = ˆ ρ = N k= 1 P k k k quilibrium 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 P ln(1/ ) k 1 k P = k and the (internal) energy N U : ( t) Hˆ ( t) P N = Ψ Ψ = k = 1 k k
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 nsemble for the populations that is Sample Space D + probability density p( P) for the populations P = ( P1, P2,, P N ) Sample space: simplex in dimensions Average of f ( P) on the nsemble: = 1 1 P 1 P P ( N ) N Pk = 1 No a priori information on p( P)! k= 1 [ ( ) ( )] 1 1 N 2 f dp dp dp f P p P 1 2 N 1 PN = 1 P P 1 N 1
3) What populations? From the lack of knowledge Random Pure State nsemble (RPS): homogeneous distribution of on the unit sphere in H N Ψ() z = Ψ() x y x, y, z = real or imaginary parts of c () k
3) What populations? From the (uclidean) measure in : N H ( N p( P) ( N 1)! 1 P ) k = δ k = 1 Note: RPS is defined only in a finite dimensional Hilbert space necessity of the high energy cut-off max From p( P) : statistical sampling of the populations and of the system s (equilibrium) properties
3) What populations? xample for randomly perturbed harmonic oscillators 8 7-4.5-5 n = number of oscillators p ( µ ) 1,1 6 5 4 8-5.5 σ µ1,1-6 -6.5-7 ( ) 2 σ f = f f = variance of f ( P) 3 2 7-7.5 5 6 7 8 n 1 n = 5 6.31.315.32.325.33.335.34.345.35.355.36 µ 1,1
3) What populations? Different realizations of the system lead to different values of the equilibrium properties, in particular of the ntropy and of the Internal nergy What a relation with Thermodynamical properties? Note: standard microcanonical analysis corresponds to a specific choice of the populations µ C µ C = 1/ N for min < k Pk = otherwise max
4) Typicality Concept of typicality from Information Theory. A is a typical event in a broad meaning if { A} { A} i) Prob = 1 ε with ε <<1, that is Prob = ε Strong form of typicality if { A} ii) Prob = 1 like the measure/probability of irrational numbers in the interval { µ 11 µ 11 σ } 11 In our case: µ i) Pro b 2.99 { µ 11 µ 11 } ii) Lim n Prob K = 1 K > [,1] 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 σ ntropy Distribution for systems of n spins 1-1 1-2 n =14 n =15 1-3 1-4 5 1 15 n n =1 n =11 n =12 n =13.645.65.655.66.665 ntropy 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 nsemble averages S U ˆ µ
5) Thermodynamics Is the RPS 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( ) = ( ) the following items are derived k δ a) Typicality in the thermodynamic limit (number of components n ) which allows the identification of the entropy ( S ), of the internal energy ( ) and of the properties of a component trough U ˆµ b) ntropy equation of state S = S( U ), by eliminating the dependence on the energy cut-off max S = f U = g S = f g U 1 ( max ) ( max ) ( ( )) k
5) Thermodynamics c) Both S and U are extensive properties therefore the Temperature 1 d S : = T d U is an intensive parameter 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 { ˆ } µ exp H / k T Hˆ : Hamiltonian of the subsystem ˆ ss B ss
6) Conclusions An open issue 1) There are other possible Statistical nsembles for the populations. In the past the Fixed xpectation nergy nsemble (F) having the constraint < Ψ Hˆ Ψ >= = 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 nsembles. As a matter of fact, F does not satisfy conditions d) and e)! 3) Further conditions to be satisfied?
ψ S 1 spin + 9 spins ( ) = k ψ k= 1 ( ) ( ) ψ () = ψ ψ ψ ( ) ( ) = S β from RPS P e max iα k Switch on the interaction Hamiltonian k M H = H1 + H + λh S Solve the Schrödinger equation Thermalization 5-5 M M z z max () H = ( Z ( )) () H = = Tr S µ =.5 S Mp z -.5 -.1 -.15 -.2 -.25 -.3 -.35 -.4 -.45 -.5 5 1 15 t
5 Thermalization 1 2 ( ).34 i α iα = +.66 ψ S e β e α M S z () H =.16 S ψ M ( ) z from RPS () H =.66 max -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 Mp zz.1 -.1 -.2 -.3 -.4 -.5 5 1 15 t