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Computation of transition trajectories and rare events in nonequilibrium statistical physics Lyon, June 11th June 15th, 2012 Universality of large deviations in diffusive systems Frdric van Wijland Laboratoire Matire et Systmes

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Lyon, June 11th – June 15th, 2012

Computation of transition trajectories and rare events in nonequilibrium statistical physics

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13/06/2012

  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations in diffusive systems

Frédéric van Wijland

Laboratoire Matière et Systèmes Complexes Université Paris Diderot – Paris 7

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13/06/2012

  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

In collaboration with

Cécile Appert, Orsay Bernard Derrida, Paris Juan P Garrahan, Nottingham Alberto Imparato, Århus Vivien Lecomte, Paris

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 1

Simple symmetric exclusion process

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 1

Simple symmetric exclusion process

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems

Coarse grained description Space scale of the system size Time scale of the diffusiion time Make occupation numbers a smoothly varying field :

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : equilibrium statics

In equilibrium, the probability to observe a given profile is

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : dynamics

Continuity equation where, on phenomenological grounds

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : dynamics

Continuity equation where, on phenomenological grounds

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : dynamics

Continuity equation where, on phenomenological grounds

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : dynamics

Green – Kubo relation

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Independent oscillators (Giardiná, Kurchan, Redig version of the

Kipnis-Marchioro-Presutti model)

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Independent oscillators

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Independent oscillators with a local conservation law for the energy

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Connection to a thermostat

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Local energy conservation

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Working at macroscopic scales

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : example 2

Equilibrium Consistent with naive continuum limit from the dynamics

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Diffusive systems : summary

Ingredients : Smooth variations over space and time scales Diffusive dynamics characterized by Noise is vanishingly small (note the )

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Fluctuations of the current (particles, energy)

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response Asking the question reveals the structure

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response Asking the question reveals the structure Evans-Morris-Cohen-Searles-Gallavotti-Cohen- Kurchan-Lebowitz-Spohn-Maes fluctuation theorem Deep understanding of the kinetic foundations of the second law

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Equilibrium statistical mechanics/thermodynamics Partial view, ignoring dynamics Nonequilibrium : quest for generic properties Mimick the approach from the statics (ideas from Ruelle & Bowen, 1970's).

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Equilibrium statistical mechanics/thermodynamics Partial view, ignoring dynamics Nonequilibrium : quest for generic properties Mimick the approach from the statics (ideas from Ruelle & Bowen, 1970's).

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

For a time realization with a prescribed current (or whatever else is of physical interest), what does the system look like ? In or out of equilibrium

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Extensive constraints : Time and space integrated current : Time and space integrated activity (traffic for C. Maes)

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Questions about rare events

Generating functions (canonical approach) : Dynamical versions of a partition function and of an intensive free energy.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique

Langevin equation for Equivalent to a path-integral formulation

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique

Action encodes dynamics + reweighting.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique

Action encodes dynamics + reweighting.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique

Partition function dominated by the saddle

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique : credits

Macroscopic fluctuation theory : Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim Lebowitz, Speer, Derrida, Bodineau See the lecture by Bernard Derrida

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique : recipe

Step 1: Find the saddle :

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

WKB-like technique : recipe

Step 2: Evaluate action at the saddle Step 3 : Integrate out fluctuations around the saddle and get

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13/06/2012

  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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13/06/2012

  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations

Consider the integrated current large deviations

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations : equilibrium

With periodic boundary conditions: Saddle is stationary and homogeneous at

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations : equilibrium

With periodic boundary conditions: where the universal scaling function has a branch cut along the real axis for

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations : equilibrium

For the SSEP, hence Burnett coefficients are infinite.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations : equilibrium

For the KMP, the argument of the scaling function Can hit the singularity : phase transition.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Universality of large deviations : equilibrium

For the KMP, the argument of the scaling function Can hit the singularity : phase transition.

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13/06/2012

  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Out of equilibrium

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Out of equilibrium

Boundary-driven diffusive systems Can be understood in terms of an additivity principle. For the SSEP (Derrida, Douçot, Roche):

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Out of equilibrium : some universality

For a constant diffusion constant and a quadratic noise strength

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Out of equilibrium : some universality

Same scaling function as in equiibrium Out of equilibrium=equilibrium in disguise For the SSEP and KMP, can be proved directly on the lattice models.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Outline

Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

Focus on the equilibrium SSEP with pbc at Activity = number of hops,

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Mott Insulator vs Superfluid in cold atoms

Anisotropy parameter

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Mott Insulator vs Superfluid in cold atoms

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

Cumulant generating function of the activity But the singularity can be hit when

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

Rephrase in terms of In the regime where

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

Saddle point equation, integrated once:

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

Saddle point equation, integrated once: Use the intuitive correspondence And search for periodic trajectories.

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Large deviations of the activity

After a few liters of sweat, at half-filling: where

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Physical picture of the phase transition

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Physical picture of the phase transition

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Physical picture of the phase transition

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  • F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

Final remarks/wishes

Large deviation issues

  • can they be measured ? See Nemoto & Sasa.

Universality issues :

  • how universal out of equilibrium ?
  • how deep is the eq/out of eq connection?

Phase transitions

  • can they be observed ?
  • can first order transitions be worked out?