Why do irreversible processes converge faster to equilibrium than reversible ones? Marcus Kaiser SAMBa Summer Conference 2017 University of Bath joint work with R. L. Jack and J. Zimmer Marcus Kaiser
Claim: Irreversible systems converge faster to equilibrium [Hwang et al. 2005][Pavliotis2013][ReyBellet-Spiliopoulos2015,2016] [Bierkens2015] Marcus Kaiser
Claim: Irreversible systems converge faster to equilibrium [Hwang et al. 2005][Pavliotis2013][ReyBellet-Spiliopoulos2015,2016] [Bierkens2015] Interesting for two reasons: • Understanding the physics of non-equilibrium systems • Acceleration of sampling methods like MCMC Marcus Kaiser
Claim: Irreversible systems converge faster to equilibrium [Hwang et al. 2005][Pavliotis2013][ReyBellet-Spiliopoulos2015,2016] [Bierkens2015] Interesting for two reasons: • Understanding the physics of non-equilibrium systems • Acceleration of sampling methods like MCMC We investigate the effect of breaking detailed balance on the convergence to the steady state. We will consider (interacting) particle systems and their hydrodynamic scaling limits. Marcus Kaiser
We consider systems on two scales: (1) Microscopic systems finite state, ergodic and irreducible continuous time Markov processes with unique steady state π and dynamics given by � µ t ( x ) = ˙ µ t ( y ) c ( y → x ) − µ t ( x ) c ( x → y ) y = L † µ t ( x ) . (2) Macroscopic systems drift diffusive systems of the form � � � � ∂ t ρ = ∇ · D ( ρ ) ∇ ρ − ∇ · χ ( ρ ) E . Marcus Kaiser
Microscopic systems Marcus Kaiser
Microscopic systems = particle systems We consider a system of indistinguishable particles which hop between sites i +1 i leading to a transition from state x to state y c ( x, y ) x y Marcus Kaiser
Microscopic systems Relations to physics: Equilibrium systems are characterised by ‘detailed balance’ π ( x ) c ( x → y ) = π ( y ) c ( y → x ) , which correspond to vanishing currents in the steady state, whereas non-equilibrium systems are characterised by a non-zero current in the steady state. The microscopic current for a measure µ is given by J x,y ( µ ) = µ ( x ) c ( x → y ) − µ ( y ) c ( y → x ) . J x,y ( π ) = 0 (for all x, y ) if and only if the system is an equilibrium system (i.e. satisfies detailed balance). Marcus Kaiser
Microscopic systems Alternative characterisation in terms of the generator L : The process is reversible (satisfies detailed balance) if L is symmetric w.r.t. the inner product in L 2 ( π ) . In general, we can write any generator L as L = L S + L A , where L S is symmetric and L A is anti-symmetric (w.r.t. L 2 ( π ) ). L S is again a generator with unique stationary measure π . Marcus Kaiser
Example We consider a system of independent particles in a potential U in 2d. 5 140 120 5 4 2.5 3.5 2 4 3.5 3 3 4.5 5 100 1.5 4.5 4 3 . 5 2.5 4 5 4 1 V(x 1 ,1/2) 80 -0.5 5 2 x 2 0 2 . 1 3 . 5 3 60 1.5 0.5 0 2.5 0 2.5 5 2 1 . 1 1 3.5 2 5 3.5 40 2.5 3 3 3.5 2 3 4 4 4.5 0 4.5 5 20 4 . 4 5 5 -1 0 20 40 60 80 100 120 140 20 40 60 80 100 120 140 x 1 x 1 We can think here of a Monte Carlo sampling with many ( ≈ 150000 ) samples. Sampling from π ∝ e − U . Lattice size L 2 = 140 × 140 . Marcus Kaiser
Example - Test observable 9 2 reversible irreversible 1.5 asymptotic value 8.5 S ( t ) 8 1 reversible irreversible 7.5 0.5 0 5 10 15 20 0 5 10 15 20 t 2 0.3 reversible reversible 0.2 irreversible average x 1 pos irreversible 1.5 0.1 D ( t ) 1 0 0.5 -0.1 0 -0.2 0 5 10 15 20 0 5 10 15 20 t t [K., Jack, Zimmer, J Stat Phys 2017] Marcus Kaiser
Acceleration of convergence ⇒ The Markov chain with generator L = L S + L A converges faster to π than the process with generator L S . This convergence can be checked in different ways: (e.g.) • The spectral gap of the generator (the largest non-zero eigenvalue of L ). • The large deviation rate functional Marcus Kaiser
Spectral gap The spectrum σ ( L ) is contained in C − := { z ∈ C | Re( z ) ≤ 0 } and 0 ∈ σ ( L ) . We denote with α ( L ) the modulus of the real part of the non-zero eigenvalue with largest real part. σ ( L S ) reversible α ( L S ) σ ( L ) α ( L ) irreversible Marcus Kaiser
Spectral gap We assume that L is diagonalisable such that we can write any distribution at time t ∈ [0 , ∞ ) as µ t ( x ) = π ( x ) + e − tα ( L ) γ ( t, x ) for a (in t ) bounded function γ ( t, x ) . Therefore � µ t − π � ≤ C e − tα ( L ) . (The initial distribution is here given by µ 0 = π + γ (0 , · ) ) Hence Theorem (Spectral gap) α ( L ) ≥ α ( L S ) . Marcus Kaiser
Large deviations Large deviations characterise asymptotic probabilities (here as t → ∞ ) in terms of a rate functional I ( µ ) . In this case, we consider the empirical � t average Θ t := 1 0 δ X u du , which satisfies t P [Θ t ≈ µ ] ≍ e − tI ( µ ) . This notation stands for the following two inequalities: For all closed sets A and open sets O , we have 1 lim sup log P [Θ t ∈ A ] ≤ − inf µ ∈ A I ( µ ) t t →∞ and 1 lim inf log P [Θ t ∈ O ] ≥ − inf µ ∈ O I ( µ ) . t →∞ t Marcus Kaiser
Large deviations We compare P [Θ t ( L S ) ≈ µ ] ≍ e − tI S ( µ ) P [Θ t ( L ) ≈ µ ] ≍ e − tI ( µ ) . and Consistently with the above result, we have Theorem (Rate functional) I S ( µ ) ≤ I ( µ ) Informally this implies that asymptotically as t → ∞ P [Θ t ( L S ) ≈ µ ] ≥ P [Θ t ( L ) ≈ µ ] for µ � = π . Marcus Kaiser
Macroscopic systems Marcus Kaiser
Macroscopic systems With the appropriate rescaling of the rates, the systems becomes on large enough scales (for large L ) ‘independent’ of the lattice size. L=150, rev. L=300, rev. 1.5 L=450, rev. L=150, irrev. D ( t ) 1 L=300, irrev. L=450, irrev. 0.5 0 0.5 1 1.5 2 t Plot of 1d system with L = 150 , 300 , 450 . The system then can be approximately described by a deterministic mass evolution. Marcus Kaiser
Macroscopic systems The macroscopic behaviour can be described in terms of a conservation law of the form ∂ t ρ t = −∇ · j t (1) for some current j t on a given domain Λ with a suitable boundary condition on ∂ Λ . Marcus Kaiser
Macroscopic systems The macroscopic behaviour can be described in terms of a conservation law of the form ∂ t ρ t = −∇ · j t (1) for some current j t on a given domain Λ with a suitable boundary condition on ∂ Λ . E.g. a box with periodic boundary condition ∂ t ρ = −∇ · j t on Λ Marcus Kaiser
Macroscopic systems The macroscopic behaviour can be described in terms of a conservation law of the form ∂ t ρ t = −∇ · j t (1) for some current j t on a given domain Λ with a suitable boundary condition on ∂ Λ . E.g. a box with periodic boundary condition ∂ t ρ = −∇ · j t on Λ For the hydrodynamic limit, the associated hydrodynamic current J ( ρ t ) is given by J ( ρ t ) = − D ( ρ t ) ∇ ρ t + χ ( ρ t ) E. (2) We assume that equation (1) with j t = J ( ρ t ) as in (2) has a unique steady state ¯ ρ . Marcus Kaiser
Splitting the current A fundamental result from the Macroscopic Fluctuation Theory (MFT) is that one can split the current in the sum of a symmetric and an anti-symmetric term: J = J S + J A which satisfies an orthogonality condition � J S ( ρ ) · χ ( ρ ) − 1 J A ( ρ ) dx = 0 . � J S ( ρ ) , J A ( ρ ) � χ ( ρ ) − 1 := Λ J S and J A can be obtained from the current of the adjoint process as J S = ( J + J ∗ ) / 2 and J A = ( J − J ∗ ) / 2 . Note: In general J S ( ρ t ) is not − D ( ρ t ) ∇ ρ t . Marcus Kaiser
Non-equilibrium systems Non-equilibrium systems correspond to the case when J A does not vanish, whereas equilibrium systems are characterised by J = J S . Similar to the microscopic case, where L = L S for reversible systems. Marcus Kaiser
Non-equilibrium systems Non-equilibrium systems correspond to the case when J A does not vanish, whereas equilibrium systems are characterised by J = J S . Similar to the microscopic case, where L = L S for reversible systems. In general, we can write the symmetric part of the current as J S ( ρ t ) = − χ ( ρ t ) ∇ δ V . δρ t where V is the so called quasipotential. • V ( ρ ) ≥ 0 with equality if and only if ρ = ¯ ρ . • V ( ρ t ) is monotonic decreasing. • V can be thought of as a (non-equilibrium) free energy that drives the system to the unique and globally attractive steady state ¯ ρ . Marcus Kaiser
Symmetric current In the case that J A ( ρ ) = 0 , we can write the dynamics as the gradient flow (or steepest descent ) ∂ t ρ t = ∇ · χ ( ρ t ) ∇ δ V . δρ t V ρ 0 ¯ ρ Recall that a gradient flow consists of a metric M and an energy V , such that ∂ t ρ t = − M ( ρ t ) δ V δρt . (Here M ( ρ t ) = −∇ · χ ( ρ t ) ∇ ). Marcus Kaiser
Example of a symmetric process Consider the linear equation ∂ t ρ t = ∆ ρ t + ∇ · ( ρ t ∇ U ) . This is the linear case (where χ ( ρ t ) = ρ t ) and the external force is of gradient type ( E = −∇ U ). Marcus Kaiser
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