Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard Vadim Oganesyan - Yale UMASS, Boston, 09/24/2008

Plan of the talk 1.Thermalization in isolated systems. 2.Connection of quantum and thermodynamic adiabatic theorems: three regimes of adiabaticity. 3.Microscopic expression for the heat and the diagonal entropy. Laws of thermodynamics and reversibility. Numerical example. 4.Expansion of quantum dynamics around the classical limit.

Cold atoms: example of isolated systems with tunable interactions. M. Greiner et. al. 2002

Ergodic Hypothesis: In sufficiently complicated systems (with stationary external parameters) time average is equivalent to ensemble average.

In the continuum this system is equivalent to an integrable KdV equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ).

Qauntum Newton Craddle. (collisions in 1D interecating Bose gas – Lieb-Liniger model) T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006) No thermalization in1D. Fast thermalization in 3D. Quantum analogue of the Fermi-Pasta- Ulam problem.

Thermalization in Quantum systems. Consider the time average of a certain observable A in an isolated system after a quench. Eignestate thermalization hypothesis (M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854, 2008.): A n,n ~ const (n) so there is no dependence on density matrix as long as it is sufficiently narrow. Necessary assumption: Information about equilibrium is fully contained in diagonal elements of the density matrix.

This is true for all thermodynamic observables: energy, pressure, magnetization, …. (pick your favorite). They all are linear in . This is not true about von Neumann entropy! Off-diagonal elements do not average to zero. The usual way around: coarse-grain density matrix (remove by hand fast oscillating off-diagonal elements of . Problem: not a unique procedure, explicitly violates time reversibility and Hamiltonian dynamics.

Von Neumann entropy: always conserved in time (in isolated systems). More generally it is invariant under arbitrary unitary transfomations Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise it increases. Quantum mechanics: for adiabatic processes there are no transitions between energy levels: If these two adiabatic theorems are related then the entropy should only depend on  nn.

Thermodynamic adiabatic theorem. General expectation: In a cyclic adiabatic process the energy of the system does not change: no work done on the system, no heating, and no entropy is generated. - is the rate of change of external parameter.

Adiabatic theorem in quantum mechanics Landau Zener process: In the limit  0 transitions between different energy levels are suppressed. This, for example, implies reversibility (no work done) in a cyclic process.

Adiabatic theorem in QM suggests adiabatic theorem in thermodynamics: Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions). 1.Transitions are unavoidable in large gapless systems. 2.Phase space available for these transitions decreases with the rate  Hence expect Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibrium

Three regimes of response to the slow ramp: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A.Mean field (analytic) – high dimensions: B.Non-analytic – low dimensions C.Non-adiabatic – low dimensions, bosonic excitations In all three situations (even C) quantum and thermodynamic adiabatic theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics.

Numerical verification (bosons on a lattice). Use the fact that quantum fluctuations are weak in the SF phase and expand dynamics in the effective Planck’s constant: Nonintegrable model in all spatial dimensions, expect thermalization.

T=0.02 Heat per site

2D, T=0.2 Heat per site

Thermalization at long times (1D).

Connection between two adiabatic theorems allows us to define heat. Consider an arbitrary dynamical process and work in the instantaneous energy basis (adiabatic basis). Adiabatic energy is the function of the state.Adiabatic energy is the function of the state. Heat is the function of the process.Heat is the function of the process. Heat vanishes in the adiabatic limit. Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!Heat vanishes in the adiabatic limit. Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!

Isolated systems. Initial stationary state. Unitarity of the evolution gives Transition probabilities p m->n are non-negative numbers satisfying In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is).

yields If there is a detailed balance then Heat is non-negative for cyclic processes if the initial density matrix is passive. Second law of thermodynamics in Thompson (Kelvin’s form). The statement is also true without the detailed balance but the proof is more complicated (Thirring, Quantum Mathematical Physics, Springer 1999).

What about entropy? Entropy should be related to heat (energy), which knows only about  nn.Entropy should be related to heat (energy), which knows only about  nn. Entropy does not change in the adiabatic limit, so it should depend only on  nn.Entropy does not change in the adiabatic limit, so it should depend only on  nn. Ergodic hypothesis requires that all thermodynamic quantities (including entropy) should depend only on  nn.Ergodic hypothesis requires that all thermodynamic quantities (including entropy) should depend only on  nn. In thermal equilibrium the statistical entropy should coincide with the von Neumann’s entropy:In thermal equilibrium the statistical entropy should coincide with the von Neumann’s entropy: Simple resolution: diagonal entropy the sum is taken in the instantaneous energy basis.

Properties of d-entropy. Jensen’s inequality: Therefore if the initial density matrix is stationary (diagonal) then Now assume that the initial state is thermal equilibrium Let us consider an infinitesimal change of the system and compute energy and entropy change.

Recover the first law of thermodynamics. If stands for the volume the we find

Classical systems. probability to occupy an orbit with energy E. Instead of energy levels we have orbits. describes the motion on this orbits. Classical d-entropy The entropy “knows” only about conserved quantities, everything else is irrelevant for thermodynamics! S d satisfies laws of thermodynamics, unlike the usually defined

Example Cartoon BCS model: Mapping to spin model (Anderson, 1958) In the thermodynamic limit this model has a transition to superconductor (XY-ferromagnet) at g = 1.

Change g from g 1 to g 2. Work with large N.

Simulations: N=2000

Entropy and reversibility.  g =  g = 10 -5

Expansion of quantum dynamics around classical limit. Classical (saddle point) limit: (i) Newtonian equations for particles, (ii) Gross-Pitaevskii equations for matter waves, (iii) Maxwell equations for classical e/m waves and charged particles, (iv) Bloch equations for classical rotators, etc. Questions: What shall we do with equations of motion? What shall we do with initial conditions? Challenge : How to reconcile exponential complexity of quantum many body systems and power law complexity of classical systems?

Partial answers. Leading order in  : equations of motion do not change. Initial conditions are described by a Wigner “probability’’ distribution: G.S. of a harmonic oscillator: Quantum-classical correspondence: ;

Semiclassical (truncated Wigner approximation): Exact for harmonic theories!Exact for harmonic theories! Not limited to low temperatures and to 1D!Not limited to low temperatures and to 1D! Asymptotically exact at short times.Asymptotically exact at short times.Summary: Expectation value is substituted by the average over the initial conditions.

Beyond the semiclassical approximation. Quantum jump. Each jump carries an extra factor of  2. Recover sign problem = exponential complexity in exact formulation of quantum dynamics.

Example (back to FPU problem). with V. Oganesyan and S. Girvin m = 10,  = 1, = 0.2, L = 100 Choose initial state corresponding to initial displacement at wave vector k = 2  /L (first excited mode). Follow the energy in the first excited mode as a function of time.

Classical simulation

Semiclassical simulation

Similar problem with bosons in an optical lattice. Prepare and release a system of bosons from a single site. Little evidence of thermalization in the classical limit. Strong evidence of thermalization in the quantum and semiclassical limits.

Many-site generalization 60 sites, populate each 10 th site.

Conclusions 1.Adiabatic theorems in quantum mechanics and thermodynamics are directly connected. 2.Diagonal entropy satisfies laws of thermodynamics from microscopics. Heat and entropy change result from the transitions between microscopic energy levels. 3.Maximum entropy state with  nn =const is the natural attractor of the Hamiltonian dynamics. 4.Exact time reversibility results in entropy decrease in time. But this decrease is very fragile and sensitive to tiny perturbations.

Illustration: Sine-Grodon model, β plays the role of  V(t) = 0.1 tanh (0.2 t)