Microscopic diagonal entropy, heat, and laws of thermodynamics
Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University R. Barankov, C. De Grandi BU V. Gritsev Fribourg, AFOSR ITAMP , 01/26/2009 Phase Diagram of One-Dimensional Bosons in Disordered Potential. E. Altman, Y. Kafri, A.P., G. Refael, PRL 2004, 2008 Coarse-grain the system n n U 2n Effective U decreases: U U 2
2 2 2 2 Remaining J decrease, 2 distribution of n 2 becomes wide Two possible scenarios: 1. U flows to zero faster than J: superfluid phase, n does not matter 2. J flows to zero faster than U: insulating phase, distribution of n determines the properties of the insulating phase Critical properties are the same for all possible filling factors!
Plan of the talk 1. Thermalization in isolated systems. 2. Connection of quantum and thermodynamic adiabatic theorems . 3. Microscopic expressions for the heat and the diagonal entropy. Laws of thermodynamics and reversibility. Thermalization in Quantum systems. Consider the time average of a certain observable A in an isolated system after a quench. A t n ,m (t ) Am,n n ,m (0)e i ( Em En )t Am,n n ,m T 1 A T
0 n ,m A(t ) dt n ,n An ,n n ,m Eignestate thermalization hypothesis (M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 , 2008.): An,n~ const (n) so there is no dependence on density matrix as long as it is sufficiently narrow. Necessary assumption: Am ,n 0, | En Em | 1 / Information about equilibrium is fully contained in diagonal elements of the density matrix. 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! S n Tr ( ln ) 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 S n (t ) Tr (t ) ln (t ) TrU U ln U U Tr ln 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: nn (t ) const (t ) If these two adiabatic theorems are related then the entropy
should only depend on nn. Thermodynamic adiabatic theorem. 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 . General expectation: E ( ) E 0 2 , S ( ) S (0) 2 - 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:
1. Transitions are unavoidable in large gapless systems. 2. Phase space available for these transitions decreases with the rateHence expect E ( ) E 0 2 , S ( ) S (0) 2 Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibrium Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions). Three regimes of response to the slow ramp: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A. Mean field (analytic) high dimensions: E ( ) E 0 2 B. Non-analytic low dimensions E ( ) E 0 | | r , r 2 C. Non-adiabatic low dimensions, bosonic excitations r
E ( ) E 0 | | L , r 2, 0 In all three situations quantum and thermodynamic adiabatic theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics. Examples Ramping in generic gapless regime E E* dE * dE * 2 2 ~ ( E*) ~ ( E*) dt d
Uniform system: E ( ) c( )k z d ln c E* ~ d E* d/z n ~ ( E ) dE ~|
| Density of quasi-particles (entropy): ex 0 Absorbed energy density (heating): Q ~| |( d z ) / z 2 all energy scales participate! n , Q ~ ( ) High dimensions: ex Adiabatic crossing quantum critical points. Relevant for adiabatic quantum
computation, adiabatic preparation of correlated states. V t, 0 How does the number of excitations (entropy, energy) scale with ? d d A.P. 2003, z 1 nex , 2 Zurek, Dorner, Zoller 2005 z 1 d Nontrivial power corresponds 2 nex ,
2 to nonlinear response! z 1 d 2( z 1 / ) is analogous to the upper critical dimension. Adiabatic nonlinear probes of 1D interacting bosons. (C. De Grandi, R. Barankov, A.P., Phys. Rev. Lett. 101, 230402, 2008) 2 K K Luttinger liquid parameter 2 K Relevant sine Gordon model: 1 2 1 2 H dx x V cos( ) 2
2 K 2 K 2 nex d/z nex 1 /( 3 K ) K=2 corresponds to a SF-IN transition in an infinitesimal lattice (H.P. Bchler, et.al. 2003) V
1.0 Optimal 1.5 2.0 2.5 K 3.0 3.5 4.0 Connection between two adiabatic theorems allows us to define heat (A.P., Phys. Rev. Lett. 101, 220402, 2008 ). Consider an arbitrary dynamical process and work in the instantaneous energy basis (adiabatic basis).
E (t , t ) n (t ) nn (t ) n (t ) nn (0) n n n (t ) nn (t ) nn (0) Ead (t ) Q(t , t ) n Adiabatic energy is the function of the state. 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! Isolated systems. Initial stationary state. nm (0) n0 nm Unitarity of the evolution gives 0 n 0
m 0 n nn (t ) pm n (t )( ) m Transition probabilities pmn are non-negative numbers satisfying p m m n (t ) pn m (t ) m In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is). nn (t ) n0 pm n (t )( m0 n0 )
m yields Q(t ) n nn (t ) n ( m0 n0 ) pm n (t ) n n If there is a detailed balance then 1 Q(t ) ( n m )( m0 n0 ) pm n (t ) 2 n Heat is non-negative for cyclic processes if the initial density 0 0 matrix is passive ( n m )( m n ) 0 . Second law of thermodynamics in Thompson (Kelvins 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 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. In thermal equilibrium the statistical entropy should coincide with the von Neumanns entropy:
1 S n Tr ( ln ) n ln n , n exp[ En ] Z n Simple resolution: diagonal entropy S d nn ln nn n the sum is taken in the instantaneous energy basis. Properties of d-entropy (R. Barankov, A. Polkovnikov, arXiv:0806.2862. ). Jensens inequality: Tr ( ln d ln d ) Tr [( d ) ln d ] 0 Therefore if the initial density matrix is stationary (diagonal) then S d (t ) S n (t ) S n (0) S d (0) Now assume that the initial state is thermal equilibrium 1
exp[ n ] Z 0 n Let us consider an infinitesimal change of the system and compute energy and entropy change. 0 n 0 n E n nn n 1 S d nn (ln 1) n0 n T n n 0
n Recover the first law of thermodynamics. E E TS d S d If stands for the volume the we find E PV TS d Classical systems. Instead of energy levels we have orbits. nn probability to occupy an orbit with energy E.
nm exp[i ( n m )t ] describes the motion on this orbits. Classical d-entropy S d N ( ) ( ) ln ( )d ( ) d ( p, q ) ( ( p, q)) The entropy knows only about conserved quantities, everything else is irrelevant for thermodynamics! Sd satisfies laws of thermodynamics, unlike the usually defined S ln . Classic example: freely expanding gas Suddenly remove the wall S Gibbs 0
by Liouville theorem 1 nn (0) nn (0 ) 2 double number of occupied states S d N ln 2 result of Hamiltonian dynamics! Example Cartoon BCS model: g H c ck 2N k k k
c k c k c p c p k,p k 1 Mapping to spin model (Anderson, 1958) k 1 g H ( S1z S 2 z ) S1 S 2 S 2 S1 2N In the thermodynamic limit this model has a transition to superconductor (XY-ferromagnet) at g = 1. Change g from g1 to g2. 0.46 N
Tr N Magnetization 0.44 Work with large N. 0.42 0.40 0.38 0 50 100 150 Time
full coarse grained max entropy 2 0 0 10 20 30 Cycle 40 50 Entropy and reversibility.
g = 10-4 g = 10-5 Conclusions 1. Adiabatic theorems in quantum mechanics and thermodynamics are directly connected. 2. Diagonal entropy S n nn ln nn 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. Probing quasi-particle statistics in nonlinear dynamical probes. (R. Barankov, C. De Grandi, V. Gritsev, A. Polkovnikov, work in progress.) T
nex T>0 T=0 1/ 3 LT Less adiabatic nex 1/ 3 ln L More adiabatic nex 1/ 2 fermionic-like 1
transition? massive fermions (hard core bosons) 0 bosonic-like massive bosons nex T K Numerical verification (bosons on a lattice). U (t ) U 0 tanh(t ) Nonintegrable model in all spatial dimensions, expect thermalization. Use the fact that quantum fluctuations are weak in the SF phase and expand dynamics in the effective Plancks constant: U / n0 J
Heat per site T=0.02 4/3 E T L 1/ 3 Heat per site 2D, T=0.2 1/ 3 E T L 1/ 3
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