Overview

1
Overview

• Intro-duction and electo-duction
• Nonequilibrium Statistical Mechanics
• A-Symetric Exclusion Process (ASEP)
• Ohms law
• Surface vs. boundary driven process
• Chemical Potentials
• Equivalence of Ensembles
• Ficks law
• Heat conduction
• Temperature gradient and heat conduction
• A more realistic conductor
• Fouriers law
• The thermoelectric effect
• Linear response theory
• Fluctuation Theorem, Green-Kubo, Onsager
  reciprocity
• Beyond linear response

2
The Girsanov formula

• The Girsanov theorem relates stochastic processes
  to white noise, from it a formula for Markov
  processes follows
• (where this sum is over jump times)

3
The time antisymmetric part

• The time-antisymmetric part of the Action (
  ) is associated to entropy
  production.
• It follows with the definition for the Girsanov
  formula that whereis the (time-integrated)
  current

4
As entropy production

• So if entropy production is associated to flows
  (currents of) energy then must be something like
  energy
• gtBoltzmann like weights that arise from the
  "local detailed balance" principle are
  reasonable, so that with ?S the entropy change
  (in the universe) we get

5
Recall for thermal conductor

• Density profile in bulk is linear analogous to 2
  bath model

G
6
Microscopic Onsager

• The heat current is proportional to the electric
  current and hence we have again Onsager
  reciprocity
• There is however a more general way to observe
  (microscopic) Onsager reciprocity

7
Thermo-Electric Effect

• Seebeck, Peltier, Thomson
• Electrons carry chargegt electric current
• Electrons interact with phonons (thermal
  vibrations of the atomic lattice)gt heat current
• gtThrough the electrons, heat and electric
  currents are coupled, but not completely!

8
Generalized model

• 2 sets of cells, 2 sets of energy levels
• Now electric field E (only atleft boundary)

E
?
E
E
9
Lattice and Statespace

• Energy levels given per wire-type
• Lattice has form
• The statespace is

10
Transformation operators

• We allow hopping with exclusion interaction in
  the same energy level
• We allow hopping between energy levels

11
Generator

• Heat exchange at the boundaries
• Exclusion in the bulk
• In total

12
Equilibrium

• Equilibrium when
• Temperatures are equal
• Electric fields are zero
• The equilibrium measure is given by
• Single cell marginal

13
Three macroscopic currents

• Microscopic (time integrated) current
• Electric current
• Heat current
• Peltier heat current

14
Conservation of Peltier current

• Energy is conserved in the bulk (due to
  conservation of particles)
• Except at the boundaries, where
• Naturally it follows that the Peltier current
  through the system is conserved (this is
  desireable since the Peltier effect is reversible)

15
Heat dissipation

• Electrical work or Joule heat
• Peltier heat exchange
• Total dissipated heat

16
Entropy production

• By construction the entropy production is
• Its average equals
• Defining eff. field, temp grad. and entr curr
• we may write

17
Macroscopic Onsager

• Fluctuation relation yields for current
• To linear order we may write
• with response coefficients

18
Back to phenomenology

• Seebeck and Peltier effects are measured in
  electrical potential and heat flow resp.
• If we define
• then Onsager reciprocity yields the Thomson
  relation

19
Beyond Linear response

• Fluctuation-dissipation thm. offers no
  resolution.
• Direct derivation of higher order response
  coefficients shows they depend on time-symmetric
  part of the action.