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

Dec 19, 2015

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Page 1: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Microscopic diagonal entropy, heat, and laws of Microscopic diagonal entropy, heat, and laws of thermodynamicsthermodynamics

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

AFOSRAFOSR

Roman Barankov – BURoman Barankov – BU

Vladimir Gritsev – HarvardVladimir Gritsev – HarvardVadim Oganesyan - YaleVadim Oganesyan - Yale

UMASS, Boston, 09/24/2008 UMASS, Boston, 09/24/2008

Page 2: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Plan of the talkPlan of the talk

1.1. Thermalization in isolated systems.Thermalization in isolated systems.

2.2. Connection of quantum and thermodynamic Connection of quantum and thermodynamic adiabatic theorems: three regimes of adiabaticity.adiabatic theorems: three regimes of adiabaticity.

3.3. Microscopic expression for the heat and the Microscopic expression for the heat and the diagonal entropy. Laws of thermodynamics and diagonal entropy. Laws of thermodynamics and reversibility. Numerical example.reversibility. Numerical example.

4.4. Expansion of quantum dynamics around the Expansion of quantum dynamics around the classical limit. classical limit.

Page 3: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Cold atoms: example of isolated systems with tunable interactions.Cold atoms: example of isolated systems with tunable interactions.

M. Greiner et. al. 2002M. Greiner et. al. 2002

Page 4: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Ergodic Hypothesis:Ergodic Hypothesis:

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

Page 5: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

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

Page 6: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

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.

Page 7: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Thermalization in Quantum systems.Thermalization in Quantum systems.

Consider the time average of a certain observable Consider the time average of a certain observable AA in an isolated in an isolated system after a quench. system after a quench.

mn

nmtEEi

mnmn

nmmn AeAttA nm

,,

)(,

,,, )0()(

mn

nnnn

TAdttA

TA

,,,0

)(1

Eignestate thermalization hypothesis (Eignestate thermalization hypothesis (M. Rigol, V. Dunjko & M. M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 , 2008Olshanii, Nature 452, 854 , 2008.): .): AAn,nn,n~ ~ const const (n) (n) so there is no so there is no

dependence on density matrix as long as it is sufficiently narrow. dependence on density matrix as long as it is sufficiently narrow.

Necessary assumption: Necessary assumption: /1||,0, mnnm EEA

Information about equilibrium is fully contained in diagonal Information about equilibrium is fully contained in diagonal elements of the density matrix.elements of the density matrix.

Page 8: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Information about equilibrium is fully contained in diagonal Information about equilibrium is fully contained in diagonal elements of the density matrix.elements of the density matrix.

This is true for all thermodynamic observables: energy, This is true for all thermodynamic observables: energy, pressure, magnetization, …. (pick your favorite). They all are pressure, magnetization, …. (pick your favorite). They all are linear in linear in ..This is not true about von Neumann entropy! This is not true about von Neumann entropy!

)ln( TrSn

Off-diagonal elements do not average to zero.Off-diagonal elements do not average to zero.

The usual way around: coarse-grain density matrix (remove by The usual way around: coarse-grain density matrix (remove by hand fast oscillating off-diagonal elements of hand fast oscillating off-diagonal elements of ..

Problem: not a unique procedure, explicitly violates time Problem: not a unique procedure, explicitly violates time reversibility and Hamiltonian dynamics.reversibility and Hamiltonian dynamics.

Page 9: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Von Neumann entropy: always conserved in time (in isolated Von Neumann entropy: always conserved in time (in isolated systems). More generally it is invariant under arbitrary unitary systems). More generally it is invariant under arbitrary unitary transfomationstransfomations

lnln)(ln)()( TrUUUTrUttTrtSn

Thermodynamics: entropy is conserved only for adiabatic Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise it increases.(slow, reversible) processes. Otherwise it increases.

Quantum mechanics: for adiabatic processes there are no Quantum mechanics: for adiabatic processes there are no transitions between energy levels: transitions between energy levels: )(const)( ttnn

If these two adiabatic theorems are related then the entropy If these two adiabatic theorems are related then the entropy should only depend on should only depend on nnnn..

Page 10: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Thermodynamic adiabatic theorem.Thermodynamic adiabatic theorem.

General expectation:General expectation:

In a cyclic adiabatic process the energy of the system In a cyclic adiabatic process the energy of the system does does not change: not change: no work done on the system, no heating, and no no work done on the system, no heating, and no entropy is generated .entropy is generated .

22 )0()(,0)( SSEE

- - is the rate of change of external parameter.is the rate of change of external parameter.

Page 11: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Adiabatic theorem in quantum mechanicsAdiabatic theorem in quantum mechanics

Landau Zener process:Landau Zener process:

In the limit In the limit 0 transitions between 0 transitions between different energy levels are suppressed.different energy levels are suppressed.

This, for example, implies reversibility (no work done) in a This, for example, implies reversibility (no work done) in a cyclic process.cyclic process.

Page 12: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Adiabatic theorem in QM Adiabatic theorem in QM suggestssuggests adiabatic theorem adiabatic theorem in thermodynamics:in thermodynamics:

Breakdown of Taylor expansion in low dimensions, Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions).especially near singularities (phase transitions).

1.1. Transitions are unavoidable in large gapless systems.Transitions are unavoidable in large gapless systems.

2.2. Phase space available for these transitions decreases with the Phase space available for these transitions decreases with the raterateHence expectHence expect

22 )0()(,0)( SSEE

Low dimensions: high density of low energy states, breakdown of Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibriummean-field approaches in equilibrium

Page 13: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Three regimes of response to the slow ramp:Three regimes of response to the slow ramp:A.P. and V.Gritsev, Nature Physics 4, 477 (2008)A.P. and V.Gritsev, Nature Physics 4, 477 (2008)

A.A. Mean field (analytic) – high dimensions: Mean field (analytic) – high dimensions:

B.B. Non-analytic – low dimensionsNon-analytic – low dimensions

C.C. Non-adiabatic – low dimensions, bosonic excitationsNon-adiabatic – low dimensions, bosonic excitations

In all three situations (even C) quantum and thermodynamic In all three situations (even C) quantum and thermodynamic adiabatic theorem are smoothly connected.adiabatic theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics.adiabatic theorem in quantum mechanics.

Page 14: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Numerical verification (bosons on a lattice).Numerical verification (bosons on a lattice).

Use the fact that quantum Use the fact that quantum fluctuations are weak in the SF fluctuations are weak in the SF phase and expand dynamics in phase and expand dynamics in the effective Planck’s constant:the effective Planck’s constant:

JnU 0/

Nonintegrable model in all spatial dimensions, expect thermalization.Nonintegrable model in all spatial dimensions, expect thermalization.

)tanh()( 0 tUtU

Page 15: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

T=0.02T=0.02

3/13/4 LTE

Hea

t per

sit

eH

eat p

er s

ite

Page 16: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

2D, T=0.22D, T=0.2

3/13/1 LTE

Hea

t per

sit

eH

eat p

er s

ite

Page 17: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Thermalization at long times (1D).Thermalization at long times (1D).

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

t=0 t=3.2/ t=12.8/ t=28.8/ t=51.2/ t=80/ Thermal

a ja

0

L/ sin(j/L)

Correlation Functions

Page 18: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Connection between two adiabatic theorems allows us to Connection between two adiabatic theorems allows us to define define heatheat..

Consider an arbitrary dynamical process and work in the Consider an arbitrary dynamical process and work in the instantaneous energy basis (adiabatic basis).instantaneous energy basis (adiabatic basis).

),()()0()()(

)0()()()(),(

tQEt

ttE

ttadn

nnnntn

nnntn

nnntnt

• 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. Heat vanishes in the adiabatic limit. Now this is not the Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!postulate, this is a consequence of the Hamiltonian dynamics!

Page 19: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Isolated systems. Initial stationary state.Isolated systems. Initial stationary state.

nmnnm 0)0(

Unitarity of the evolution givesUnitarity of the evolution gives

m

nmnmnnm tpt ))(()( 000

Transition probabilities Transition probabilities ppm->nm->n are non-negative numbers satisfying are non-negative numbers satisfying

m n

mnnm tptp )()(

In general there is no detailed balance even for cyclic In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is).processes (but within the Fremi-Golden rule there is).

Page 20: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

m

nmnmnnn tpt ))(()( 000

yieldsyields

n

nmnmnn

nnn tpttQ )()()()( 00

If there is a detailed balance thenIf there is a detailed balance then

n

nmnmmn tptQ )())((2

1)( 00

Heat is non-negative for cyclic processes if the initial density Heat is non-negative for cyclic processes if the initial density matrix is passive . matrix is passive . Second law of Second law of thermodynamics in Thompson (Kelvin’s form).thermodynamics in Thompson (Kelvin’s form).

0))(( 00 nmmn

The statement is also true without the detailed balance but the proof The statement is also true without the detailed balance but the proof is more complicated is more complicated (Thirring, Quantum Mathematical Physics, Springer (Thirring, Quantum Mathematical Physics, Springer 1999).1999).

Page 21: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

What about entropy?What about entropy?

• Entropy should be related to heat (energy), which knows Entropy should be related to heat (energy), which knows only about only about nnnn..

• Entropy does not change in the adiabatic limit, so itEntropy does not change in the adiabatic limit, so it should should depend only on depend only on nnnn..

• Ergodic hypothesis requires that all thermodynamic Ergodic hypothesis requires that all thermodynamic quantities (including entropy) should depend only on quantities (including entropy) should depend only on nnnn..

• In thermal equilibrium the statistical entropy should In thermal equilibrium the statistical entropy should coincide with the von Neumann’s entropy:coincide with the von Neumann’s entropy:

]exp[1

,ln)ln( nnn

nnn EZ

TrS Simple resolution: diagonal entropySimple resolution: diagonal entropy

n

nnnndS ln the sum is taken in the the sum is taken in the instantaneous energy basis.instantaneous energy basis.

Page 22: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Properties of d-entropy.Properties of d-entropy.

Jensen’s inequality:Jensen’s inequality:

0]ln)[()lnln( dddd TrTr

Therefore if the initial density matrix is stationary (diagonal) thenTherefore if the initial density matrix is stationary (diagonal) then

)0()0()()( dnnd SStStS

Now assume that the initial state is thermal equilibriumNow assume that the initial state is thermal equilibrium

]exp[10

nn Z

Let us consider an infinitesimal change of the system and Let us consider an infinitesimal change of the system and compute energy and entropy change.compute energy and entropy change.

Page 23: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

nnn

nnnnd

nnnnnn

TS

E

00

00

1)1(ln

Recover the first law of thermodynamics.Recover the first law of thermodynamics.

dS

STE

Ed

If If stands for the stands for thevolume the we findvolume the we find dSTVPE

Page 24: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Classical systems.Classical systems.

nn probability to occupy an orbit with energy E.probability to occupy an orbit with energy E.

Instead of energy levels we Instead of energy levels we have orbits.have orbits.

])(exp[ ti mnnm describes the motion on describes the motion on this orbits. this orbits.

Classical d-entropyClassical d-entropy

dNSd )(ln)()( )),((),()( qpqpd

The entropy “knows” only about conserved quantities, The entropy “knows” only about conserved quantities, everything else is irrelevant for thermodynamics! Severything else is irrelevant for thermodynamics! Sdd satisfies satisfies

laws of thermodynamics, unlike the usually defined laws of thermodynamics, unlike the usually defined .lnS

Page 25: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

ExampleExample

Cartoon BCS model:Cartoon BCS model:

pkppkk

kkkk cccc

N

gccH

,2

1k1k

Mapping to spin model (Anderson, 1958)Mapping to spin model (Anderson, 1958)

122121 2)( SSSS

N

gSSH zz

In the thermodynamic limit this model has a transition to In the thermodynamic limit this model has a transition to superconductor (XY-ferromagnet) at superconductor (XY-ferromagnet) at g = g = 1.1.

Page 26: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Change Change gg from from gg11 to to gg22..

0 50 100 150 200 250

0.38

0.40

0.42

0.44

0.46

Mag

netiz

atio

n

Time

NT

N

r

Work with Work with large N.large N.

Page 27: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

0 10 20 30 40 50

0.38

0.40

0.42

0.44

0.46

M

agne

tizat

ion

Cycle

Full Coarse-grained

Simulations: N=2000Simulations: N=2000

Page 28: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

0 10 20 30 40 500

2

4

6

8

full coarse grained max entropy

D-E

ntr

op

y

Cycle

He

at

Page 29: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Entropy and reversibility.Entropy and reversibility.

g = g = 1010-4-4

g = g = 1010-5-5

Page 30: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Expansion of quantum dynamics around classical limit.Expansion of quantum dynamics around classical limit.

Classical (saddle point) limit: Classical (saddle point) limit:

(i) Newtonian equations for particles, (i) Newtonian equations for particles,

(ii) Gross-Pitaevskii equations for matter waves, (ii) Gross-Pitaevskii equations for matter waves,

(iii) Maxwell equations for classical e/m waves and charged particles, (iii) Maxwell equations for classical e/m waves and charged particles,

(iv) Bloch equations for classical rotators, etc.(iv) Bloch equations for classical rotators, etc.Questions:Questions:

What shall we do with equations of motion?What shall we do with equations of motion?

What shall we do with initial conditions?What shall we do with initial conditions?

Challenge : Challenge : How to reconcile exponential complexity of quantum many body How to reconcile exponential complexity of quantum many body systems and power law complexity of classical systems?systems and power law complexity of classical systems?

Page 31: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Partial answers.Partial answers.

Leading order in Leading order in : equations of motion do not change. Initial : equations of motion do not change. Initial conditions are described by a Wigner “probability’’ distribution:conditions are described by a Wigner “probability’’ distribution:

/)2/()2/(*),( ipexxdpxW

G.S. of a harmonic G.S. of a harmonic oscillator:oscillator:

Quantum-classical correspondence:Quantum-classical correspondence:

;;

Page 32: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Semiclassical (truncated Wigner approximation):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:Summary:

Expectation value is substituted by the average over the initial conditions. Expectation value is substituted by the average over the initial conditions.

Page 33: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Beyond the semiclassical approximation.Beyond the semiclassical approximation.

Quantum jump.Quantum jump.

3 p

Each jump carries an extra factor of Each jump carries an extra factor of 2.2.

Recover sign problem = exponential complexity in exact Recover sign problem = exponential complexity in exact formulation of quantum dynamics. formulation of quantum dynamics.

Page 34: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Example (back to FPU problem) .Example (back to FPU problem) .with V. Oganesyan and S. Girvinwith V. Oganesyan and S. Girvin

m m = 10, = 10, = 1, = 1, = 0.2, = 0.2, L L = 100= 100

Choose initial state corresponding to initial displacement at wave Choose initial state corresponding to initial displacement at wave vector vector k k = 2= 2//L L (first excited mode)(first excited mode)..

Follow the energy in the first excited mode as a function of time.Follow the energy in the first excited mode as a function of time.

Page 35: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Classical simulationClassical simulation

0 10000 20000 30000 40000 500000

5

10

15

20

Ene

rgy

Time

Page 36: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Semiclassical simulationSemiclassical simulation

Page 37: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

-1

-3

-5-7

-9-11

10080

6040

20

0.5

1.0

1.5

2.0

Time/1000

Ene

rgy

ln

Page 38: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

Similar problem with bosons in an optical lattice.Similar problem with bosons in an optical lattice.

Prepare and release Prepare and release a system of bosons a system of bosons from a single site.from a single site.

Little evidence of thermalization in the classical limit. Strong Little evidence of thermalization in the classical limit. Strong evidence of thermalization in the quantum and semiclassical limits.evidence of thermalization in the quantum and semiclassical limits.

Page 39: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

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

Page 40: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

ConclusionsConclusions

1.1. Adiabatic theorems in quantum mechanics and Adiabatic theorems in quantum mechanics and thermodynamics are directly connected. thermodynamics are directly connected.

2.2. Diagonal entropy satisfies laws of Diagonal entropy satisfies laws of thermodynamics from microscopics. Heat and entropy thermodynamics from microscopics. Heat and entropy change result from the transitions between microscopic change result from the transitions between microscopic energy levels.energy levels.

3.3. Maximum entropy state with Maximum entropy state with nnnn=const is the natural =const is the natural

attractor of the Hamiltonian dynamics.attractor of the Hamiltonian dynamics.

4.4. Exact time reversibility results in entropy decrease in time. Exact time reversibility results in entropy decrease in time. But this decrease is very fragile and sensitive to tiny But this decrease is very fragile and sensitive to tiny perturbations.perturbations.

n nnnnS ln

Page 41: Microscopic diagonal entropy, heat, and laws of thermodynamics Anatoli Polkovnikov, Boston University AFOSR Roman Barankov – BU Vladimir Gritsev – Harvard.

)cos(

2

1

2

1 22 VdxH x

Illustration: Sine-Grodon model, Illustration: Sine-Grodon model, ββ plays the role of plays the role of

V(t) V(t) = 0.1 tanh (0.2 = 0.1 tanh (0.2 t)t)

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10

cos

t

TWA

1st Correction Linear Response

Tonks 2

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

Semiclassical

1st Correction Linear Response

t