Exact Solutions in Non-equilibrium Statistical Physics K. Mallick Institut de Physique Th´ eorique, CEA Saclay (France) Forum de la Th´ eorie, Saclay, 3 Avril 2013 K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Introduction The statistical mechanics of a system at thermal equilibrium is encoded in the Boltzmann-Gibbs canonical law: P eq ( C ) = e − E ( C ) / kT Z the Partition Function Z being related to the Thermodynamic Free Energy F: F = − kTLog Z This provides us with a well-defined prescription to analyze systems at equilibrium : (i) Observables are mean values w.r.t. the canonical measure. (ii) Statistical Mechanics predicts fluctuations (typically Gaussian) that are out of reach of Classical Thermodynamics. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Systems far from equilibrium No fundamental theory is yet available. • What are the relevant macroscopic parameters? • Which functions describe the state of a system? • Do Universal Laws exist? Can one define Universality Classes? • Can one postulate a general form for the microscopic measure? • What do the fluctuations look like (‘non-gaussianity’)? Example: Stationary driven systems in contact with reservoirs. J R2 R1 K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Rare Events and Large Deviations Let ǫ 1 , . . . , ǫ N be N independent binary variables, ǫ k = ± 1, with probability p (resp. q = 1 − p ) . Their sum is denoted by S N = � N 1 ǫ k . • The Law of Large Numbers implies that S N / N → p − q a.s. √ • The Central Limit Theorem implies that [ S N − N ( p − q )] / N converges towards a Gaussian Law. One can show that for − 1 < r < 1, in the large N limit, � S N � ∼ e − N Φ( r ) Pr N = r where the positive function Φ( r ) vanishes for r = ( p − q ). The function Φ( r ) is a Large Deviation Function: it encodes the probability of rare events. Φ( r ) = 1 + r � 1 + r � + 1 − r � 1 − r � ln ln 2 2 p 2 2 q K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Density fluctuations in a gas Mean Density ρ 0 = N V v n In a volume v s. t. 1 ≪ v ≪ V N � n v � = ρ 0 V, T The probability of observing large fluctuations of density in v is given by � n � ∼ e − v Φ( ρ ) v = ρ Pr Φ( ρ ) = f ( ρ, T ) − f ( ρ 0 , T ) − ( ρ − ρ 0 ) ∂ f with ∂ρ 0 where f ( ρ, T ) is the free energy per unit volume in units of kT : the Thermodynamic Free Energy can be viewed as a Large Deviation Function. Conversely, large deviation functions may play the role of potentials in non-equilibrium statistical mechanics. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
A Symmetry of the Large Deviation Function Large deviation functions obey a symmetry that remains valid far from equilibrium: Φ( r ) − Φ( − r ) = Ar The coefficient A is a constant, e.g. A = ln q / p in the example above. This Fluctuation Theorem of Gallavotti and Cohen is deep and general: it reflects covariance properties under time-reversal. In the vicinity of equilibrium the Fluctuation Theorem yields the fluctuation-dissipation relation (Einstein), Onsager’s relations and linear response theory (Kubo). K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Total Current transported through a System A paradigm of a non-equilibrium system J R2 R1 K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Total Current transported through a System A paradigm of a non-equilibrium system J R2 R1 The asymmetric exclusion model with open boundaries α β q 1 RESERVOIR RESERVOIR 1 L γ δ K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Classical Transport in 1d: ASEP q p p q p Asymmetric Exclusion Process. A paradigm for non-equilibrium Statistical Mechanics. • EXCLUSION: Hard core-interaction; at most 1 particle per site. • ASYMMETRIC: External driving; breaks detailed-balance • PROCESS: Stochastic Markovian dynamics; no Hamiltonian. SOME APPLICATIONS: • Low dimensional transport. • Sequence matching, Brownian motors. • Traffic and Pedestrian flow. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
ORIGINS • Interacting Brownian Processes (Spitzer, Harris, Liggett). • Driven diffusive systems (Katz, Lebowitz and Spohn). • Transport of Macromolecules through thin vessels. Motion of RNA templates. • Hopping conductivity in solid electrolytes. • Directed Polymers in random media. Reptation models. APPLICATIONS • Traffic flow. • Sequence matching. • Brownian motors. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Elementary Model for Protein Synthesis C. T. MacDonald, J. H. Gibbs and A.C. Pipkin, Kinetics of biopolymerization on nucleic acid templates, Biopolymers (1968). K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
An Important Mathematical Result Consider the Symmetric Exclusion Process on an infinite one-dimensional lattice with spacing a and with a finite density ρ of particles. Suppose that we tag and observe a particle that was initially located at site 0 and monitor its position X t with time. On the average � X t � = 0 but how large are its fluctuations? • If the particles were non-interacting (no exclusion constraint), each particle would diffuse normally � X 2 t � = 2 Dt . K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
An Important Mathematical Result Consider the Symmetric Exclusion Process on an infinite one-dimensional lattice with spacing a and with a finite density ρ of particles. Suppose that we tag and observe a particle that was initially located at site 0 and monitor its position X t with time. On the average � X t � = 0 but how large are its fluctuations? • If the particles were non-interacting (no exclusion constraint), each particle would diffuse normally � X 2 t � = 2 Dt . • Because of the exclusion condition, a particle displays an anomalous diffusive behaviour: � t � = 21 − ρ Dt � X 2 a ρ π T.E. Harris, J. Appl. Prob. (1965). F. Spitzer, Adv. Math. (1970). K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
A crystal growing on a corner in two dimensions _ + y S t + x K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Mapping to a one-dimensional particle process y x z K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
The Hydrodynamic Limit E = ν / 2L ρ ρ 1 2 L Starting from the microscopic level, define local density ρ ( x , t ) and current j ( x , t ) with macroscopic space-time variables x = i / L , t = s / L 2 (diffusive scaling). The typical evolution of the system is given by the hydrodynamic behaviour: ∂ t ρ = 1 2 ∇ 2 ρ − ν ∇ σ ( ρ ) with σ ( ρ ) = ρ (1 − ρ ) (Lebowitz, Spohn, Varadhan) K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Large Deviations at the Hydrodynamic Level What is the probability to observe an atypical current j ( x , t ) and the corresponding density profile ρ ( x , t ) during 0 ≤ s ≤ L 2 T ? Pr { j ( x , t ) , ρ ( x , t ) } ∼ e − L I ( j ,ρ ) where the Large-Deviation functional is given by macroscopic fluctuation theory (Jona-Lasinio et al.) � T � 1 � 2 j − νσ ( ρ ) + 1 � 2 ∇ ρ I ( j , ρ ) = dt dx σ ( ρ ) 0 0 with the constraint: ∂ t ρ = −∇ . j This leads to a variational procedure to control a deviation of the density and of the associated current: an optimal path problem. This is a global framework. Unfortunately, the corresponding Euler-Lagrange equations can not be solved analytically in general. Our aim is to derive the statistical properties of the current and its large deviations starting from the microscopic model. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Current Fluctuations on a ring K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Markov Equation for the ASEP on a ring 1 x L SITES N PARTICLES ( N ) L Ω = CONFIGURATIONS x asymmetry parameter Master Equation for the Probability P t ( x 1 , . . . , x N ) of being in configuration 1 ≤ x 1 < . . . < x N ≤ L at time t . dP t ′ [ P t ( x 1 , . . . , x i − 1 , . . . , x N ) − P t ( x 1 , . . . , x i , . . . x N )] � = d t i ′ [ P t ( x 1 , . . . , x i + 1 , . . . , x N ) − P t ( x 1 , . . . , x i , . . . x N )] � + x i = MP . The sum being restricted to admissible configurations. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Large Deviations of the Current Let Y t be the total current i.e. total distance covered by all the N particles, hopping on a ring of size L , between time 0 and time t . Y t In the stationary state, a non-vanishing mean-current: t → J The fluctuations of Y t obey a Large Deviation Principle: � Y t � ∼ e − t Φ( j ) P t = j Φ( j ) being the large deviation function of the total current. Equivalently, consider the moment-generating function, which when t → ∞ , behaves as e µ Y t � ≃ e E ( µ ) t � Related by Legendre transform: E ( µ ) = max j ( µ j − Φ( j )) The calculation of E ( µ ) can be identified to eigenvalue problem solvable by Bethe Ansatz. K. Mallick Exact Solutions in Non-equilibrium Statistical Physics
Recommend
More recommend
