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
Equilibrium detailed balance

• Equilibrium for ASEP on a ring iff
• Equilibrium corresponds with detailed balance
• Equilibrium measure Gibbs measure product
  measure on canonical ensemble (density conserving
  ensemble)

3
Particle reservoirs
1
N
2

• Open the ring
• Set
• Attach particle reservoirs with
• inverse temperatures
• chemical potentials
• Possible gradient current entropy production

N-1
3
1
2
3
N
N-1
4
Chemical potential

• Imagine a very large "bath" of particles at a
  certain inverse temperature
• The chemical potential is in fact a Lagrange
  multiplyer that sets the average density of the
  bath (with respect to the system)
• For a system connected to such bath, we deal with
  the grand canonical ensmeble because the particle
  amount may change

5
The generator (dynamics)

• Interaction with particle bath
• Exclusion dynamics
• Combined generator

6
Equilibrium measure

• Equilibrium when
• The (stationary) equilibrium measure
• (excercise! check this equation (calc ZN ) show
  that it is the st. equilibrium measure)

7
The (neq.) density profile

• One site marginal/density function
• For the bulk we may derive for stationarity
• The density profile becomes
• are unknown

8
Particle Current

• Conserv. of particles density prof. yields
• (exercise! derive the formula for the current)

9
Heat Current

• Energy current into system left and right, sum
  must be dissipated, rest transport
• We may write to linear order

10
Ficks law

• Fick
• "Fourier"
• Onsager reciprocity is trivial here since the
  coupling between heat and electric current is
  one-to-one
• (exercise! demonstrate )

11
Entropy production

• Information or Shannon entropy
• Space-time anal. mean entropy production

12
Comparisson with ASEP

• In order to compare the "two-particle-bath"
  system with ASEP-on-a-ring we need that the
  densities coincide
• We need
• Thermodynamic limit
• Equivalence of ensembles (canonical and grand
  canonical)

13
Equivalence of ensembles

• Fluctuations of the density for grand canonical
  ensemble decay
• The equivalence of ensembles may in equilibrium
  (!) be expressed as
• (exercise!

14
Bulk vs. boundary driven processes

• Which is more effective/efficient?
• (exercise!
• Compare currents of 2 baths vs. ASEP for
• Which yields greatest current?
• Would you expect this is general (for boundary
  vs. bulk driven processes)?
• (heuristically) explain your results!)