Microscopic diagonal entropy, heat, and laws of thermodynamics

1
Microscopic diagonal entropy, heat, and laws of
thermodynamics
Anatoli Polkovnikov, Boston University
Roman Barankov BU Vladimir Gritsev
HarvardVadim Oganesyan - Yale
UMASS, Boston, 09/24/2008
AFOSR
2
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.

3
Cold atoms example of isolated systems with
tunable interactions.
M. Greiner et. al. 2002
4
Ergodic Hypothesis
In sufficiently complicated systems (with
stationary external parameters) time average is
equivalent to ensemble average.
5
In the continuum this system is equivalent to an
integrable KdV equation. The solution splits into
non-thermalizing solitons Kruskal and Zabusky
(1965 ).
6
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)
7
Thermalization in Quantum systems.
Consider the time average of a certain observable
A in an isolated system after a quench.
Information about equilibrium is fully contained
in diagonal elements of the density matrix.
8
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.
9
Von Neumann entropy always conserved in time (in
isolated systems). More generally it is invariant
under arbitrary unitary transfomations
If these two adiabatic theorems are related then
the entropy should only depend on ?nn.
10
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
- is the rate of change of external parameter.
11
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.
12
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 rate. Hence expect

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).
13
Three regimes of response to the slow ramp A.P.
and V.Gritsev, Nature Physics 4, 477 (2008)

1. Mean field (analytic) high dimensions
2. Non-analytic low dimensions
3. 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.
14
Numerical verification (bosons on a lattice).
Nonintegrable model in all spatial dimensions,
expect thermalization.
15
T0.02
Heat per site
16
2D, T0.2
Heat per site
17
Thermalization at long times (1D).
18
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.
• 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!

19
Isolated systems. Initial stationary state.
Unitarity of the evolution gives
Transition probabilities pm-gtn are non-negative
numbers satisfying
In general there is no detailed balance even for
cyclic processes (but within the Fremi-Golden
rule there is).
20
yields
The statement is also true without the detailed
balance but the proof is more complicated
(Thirring, Quantum Mathematical Physics, Springer
1999).
21
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

22
Properties of d-entropy.
Jensens inequality
Therefore if the initial density matrix is
stationary (diagonal) then
23
(No Transcript)
24
Classical systems.
Instead of energy levels we have orbits.
25
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.
26
Change g from g1 to g2.
Work with large N.
27
Simulations N2000
28
(No Transcript)
29
Entropy and reversibility.
?g 10-4
?g 10-5
30
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.
31
Partial answers.
Leading order in ? equations of motion do not
change. Initial conditions are described by a
Wigner probability distribution
32
Semiclassical (truncated Wigner approximation)
Expectation value is substituted by the average
over the initial conditions.
Summary
33
(No Transcript)
34
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.
35
Classical simulation
36
Semiclassical simulation
37
(No Transcript)
38
Similar problem with bosons in an optical lattice.
39
Many-site generalization 60 sites, populate each
10th site.
40
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 ?nnconst 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.

41
Illustration Sine-Grodon model, ß plays the role
of ?
V(t) 0.1 tanh (0.2 t)