Markovian Approximation and Linear Response
Theory for Classical Open Systems
G.A. Pavliotis
INRIA-MICMAC Project Team
and
Department of Mathematics Imperial College London
Lecture 1 09/11/2011
Lecture 2 28/11/2011
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Contents
The Langevin equation.
Classical open systems and the Kac-Zwanzig model.
Markovian approximation.
Derivation of the Lagevin equation.
Linear response theory for the Langevin equation.
The Green-Kubo formalism.
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Consider the Langevin equation
q̈ = −∇V (q) − γ q̇ +
p
2γβ −1 Ẇ ,
(1)
with V (q) being a confining potential (later on we will also
consider periodic potentials), γ is the friction coefficient and β the
inverse temperature.
Write it as the first order system
dqt
= pt dt,
(2a)
dpt
p
= −∇V (qt ) dt − γpt dt + 2γβ −1 dWt .
(2b)
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The process (qt , pt ) is a Markov process on Rd × Rd (or Td × Rd
for periodic potentials) with generator
L = p · ∇q − ∇q V (q) · ∇p + γ(−p · ∇p + β −1 ∆p ).
(3)
The Fokker-Planck operator (L2 (R2d )−adjoint of the generator) is
L∗ · = −p · ∇q · +∇q V · ∇p · +γ ∇p (p·) + β −1 ∆p ·
(4)
The law of the process at time t (distribution function) ρ(q, p, t) is
the solution of the Fokker-Planck equation
∂ρ
= L∗ ρ,
∂t
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ρ|t=0 = ρ0 .
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(5)
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Introduce the notation Xtq,p := qt , pt ; q0 = q, p0 = p .
The expectation of an observable
u(q, p, t) = Ef (Xtq,p ),
can be computed by solving the backward Kolmogorov equation
∂u
= Lu,
∂t
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u|t=0 = f (q, p).
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(6)
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For sufficiently nice potentials the Langevin dynamics (2) is
ergodic with respect to the Gibbs measure
µ(dqdp) =
1 −βH(q,p)
e
dqdp =: ρ∞ (q, p) dpdq,
Z
Z
Z =
e−βH(q,p) dqdp.
(7a)
(7b)
R2d
The density ρ∞ can be obtained as the solution of the stationary
Fokker-Planck equation L∗ ρ∞ = 0.
Furthermore, we have exponentially fast convergence to
equilibrium
kρ(t, ·) − ρ∞ k ≤ Ce−λt ,
(8)
for positive constants C, λ. See the lecture notes by F. Nier.
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Why use the Langevin equation?
I
I
As a thermostat, i.e. to sample from the distribution (7) and to also
compute time dependent quantities such as correlation functions.
As a reduced description of a more complicated system. Think of
the physical model of Brownian motion: colloid particle interacting
with the molecules of the surrounding fluid. We obtain a closed
(stochastic) equation for the dynamics of the Brownian particle by
”integrating out” the fluid molecules.
One of the goals of these lectures is to derive the Langevin
equation from a more basic model (”first principles”).
We also want to give a miscroscopic definition of the friction
coefficient γ.
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It is often useful (also in experimental set-ups) to probe a system
which is at equilibrium by adding a weak external forcing.
The perturbed Langevin dynamics (assumed to be at equilibrium
at t = 0) is
p
q̈ = −∇V (q ) + F (t) − γ q̇ + 2γβ −1 Ẇ .
(9)
We are interested in understanding the dynamics in the weak
forcing regime 1. We will do this by developing linear
response theory (first order perturbation theory).
We will see that the response of the system to the external forcing
is related to the fluctuations at equilibrium.
A result of this analysis is the derivation of Green-Kubo formulas
for transport coefficients. For the diffusion coefficient we have
(Einstein relation/Green-Kubo)
Z +∞
D = β −1 µ =
hpt p0 ieq dt,
(10)
0
where µ =
limF →0 VF ,
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where V is the effective drift.
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REFERENCES
K. Lindenberg and B.J. West The nonequilibrium statistical
mechanics of open and closed systems, 1990.
R.M. Mazo Brownian motion, 2002.
R. Zwanzig nonequilibrium statistical mechanics, 2001.
R. Kubo, M. Toda, N. Hashitsume Statistical Physics II.
Nonequilibrium statistical mechanics, 1991.
L. Rey-Bellet Open Classical Systems. In Open Quantum
Systems II, 2005.
D. Givon, R. Kupferman, A.M. Stuart, Extracting macroscopic
dynamics: model problems and algorithms, Nonlinearity, 17(6),
R55–R127, (2004).
Lecture notes, G.P.
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DERIVATION OF THE LANGEVIN EQUATION
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We will consider the dynamics of a Brownian particle (”small
system”) in contact with its environment (”heat bath”).
We assume that the environment is at equilibrium at time t = 0.
We assume that the dynamics of the full system is Hamiltonian:
H(Q, P; q, p) = HBP (Q, P) + H(q, p) + λHI (Q, q),
(11)
where Q, P are the position and momentum of the Brownian
particle, q, p of the environment (they are actually fields) and λ
controls the strength of the coupling between the Brownian
particle and the environment.
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Let H denote the phase space of the full dynamics and
X := (P, Q, p, q). Our goal is to show that in some appropriate
asymptotic limit we can obtain a closed equation for the dynamics
of the Brownian particle.
In other words, we need to find a projection P : H 7→ R2n so that
PX satisfies an equation that is of the Langevin type (2).
We can use projection operator techniques at the level of the
Liouville equation
∂ρ
= {H, ρ}
(12)
∂t
associated to the Hamiltonian dynamics (11)–Mori-Zwanzig
formalism.
Projection operator techniques tend to give formal results.
Additional approximation/asymptotic calculations are needed.
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We will consider a model for which we can obtain the Langevin
equation (2) from the Hamiltonian dynamics (11) in a quite explicit
way.
We will make two basic assumptions:
1
2
The coupling between the Brownian particle and the environment is
linear.
The Hamiltonian describing the environment is quadratic in
positions (and momenta).
In addition, the environment is much larger than the Brownian
particle (i.e. infinite dimensional) and at equilibrium at time t = 0.
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We have 4 different levels of description:
1
2
The full Hamiltonian dynamics (11).
The generalized Langevin equation (GLE) that we will obtain after
eliminating the heat bath variables:
Z t
Q̈ = −∇V (Q) −
γ(t − s)Q̇(s) ds + F (t).
(13)
0
3
4
The Langevin equation (1) that we will obtain in the rapid
decorrelation limit
The overdamped Langevin (Smoluchowski) dynamics that we
obtain in the high friction limit
p
q̇ = −∇V (q) + 2β −1 Ẇ .
(14)
Each reduced level of description includes less information but is
easier to analyze.
For example, we can prove exponentially fast convergence to
equilibrium for (1) and (14), something that is not known, without
additional assumptions, for (11) and (13).
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The Hamiltonian of the Brownian particle (in one dimension, for
simplicity) is described by the Hamiltonian (notice change of
notation)
1
HBP = p2 + V (q),
(15)
2
where V is a confining potential.
The environment is modeled as a linear the wave equation (with
infinite energy):
∂t2 φ(t, x) = ∂x2 φ(t, x).
(16)
The Hamiltonian of this system (which is quadratic) is
Z
HHB (φ, π) =
|∂x φ|2 + |π(x)|2 dx.
(17)
where π(x) denotes the conjugate momentum field.
The initial conditions are distributed according to the Gibbs
measure (which in this case is a Gaussian measure) at inverse
temperature β, which we formally write as
”µβ = Z −1 e−βHHB (φ,π) dφdπ”.
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Under this assumption on the initial conditions, typical
configurations of the heat bath have infinite energy. In this way,
the environment can pump enough energy into the system so that
non-trivial fluctuations emerge.
The coupling between the particle and the field is linear:
Z
HI (Q, φ) = q ∂x φ(x)ρ(x) dx,
(19)
where The function ρ(x) models the coupling between the particle
and the field.
The coupling (19) can be thought of as the first term in a Taylor
series expansion of a nonlinear, nonlocal coupling (dipole coupling
approximation).
The Hamiltonian of the particle-field model is
H(Q, P, φ, π) = HBP (P, Q) + H(φ, π) + λHI (Q, φ).
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Let us first consider a caricature of (20) where there is only one
oscillator in the ”environment”:
2
P2
p
1 2 2
H(Q, P, q, p) =
+ V (Q) +
+ ω q − λqQ , (21)
2
2
2
Hamilton’s equations of motion are:
Q̈ + V 0 (Q) = λq,
(22a)
2
(22b)
q̈ + ω (q − λQ) = 0.
We can solve (22b) using the variation of constants formula. Set
z = (q p)T , p = q̇. Eqn (22b) can be written as
dz
= Az + λh(t),
dt
where
A=
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0
1
−ω 2 0
(23)
and h(t) =
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Q(t)
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The solution of (23) is
At
Z
z(t) = e z(0) + λ
t
eA(t−s) h(s) ds.
0
We calculate
1
sin(ωt)A,
(24)
ω
Consequently: where I stands for the 2 × 2 identity matrix. From
this we obtain
eAt = cos(ωt)I +
p(0)
sin(ωt)
q(t) = q(0) cos(ωt) +
ω
Z t
1
sin(ω(t − s))Q(s) ds.
+λ
ω 0
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Now we substitute (25) into (22a) to obtain a closed equation that
describes the dynamics of the Brownian particle:
Z t
Q̈ = −V 0 (Q) − λ2
D(t − s)Q(s) ds + λF (t),
(26)
0
where
D(t) = −
1
sin(ωt),
ω
(27)
and
p(0)
sin(ωt).
(28)
ω
Since q(0) and p(0) are Gaussian random variables with mean 0
and hq(0)2 i = β −1 ω −2 , hp(0)2 i = β −1 , hq(0)p(0)i = 0 we have
F (t) = q(0) cos(ωt) +
hF (t)F (s)i = β −1 ω −2 cos(ω(t − s)) =: β −1 C(t − s).
Consequently,
D(t) = Ċ(t).
(29)
This is a form of the fluctuation-dissipation theorem.
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We perform an integration by parts in (25):
λ
p(0)
sin(ωt)
q(t) =
q(0) − 2 Q(0) cos(ωt) +
ω
ω
Z t
1
1
+λ 2 Q(t) − λ 2
cos(ω(t − s))Q̇(s) ds.
ω
ω 0
We substitute this in equation (22a) to obtain the Generalized
Langevin Equation (GLE)
Z t
2
Q̈ = −Veff (Q) − λ
γ(t − s)Q̇(s) ds + λF (t),
(30)
0
where Veff (Q) = V (Q) −
λ2
Q2,
2ω 2
1
cos(ωt),
(31a)
2
ω
h
i
λ
p(0)
F (t) =
q(0) − 2 Q(0) cos(ωt) +
sin(ωt) . (31b)
ω
ω
γ(t) =
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The same calculation can be done for an arbitrary number of
harmonic oscillators in the environment.
When writing the dynamics of the Brownian particle in the
form (30) we need to introduce an effective potential.
We have assumed that the initial conditions of the Brownian
particle are deterministic, independent of the initial distribution of
the environment. i.e. the environment is initially at equilibrium in
the absence of the Brownian particle.
We can also assume that the environment is initially in equilibrium
in the presence of the distinguished particle, i.e. that the initial
positions and momenta of the heat bath particles are distributed
according to a Gibbs distribution, conditional on the knowledge of
{Q0 , P0 }:
µβ (dpdq) = Z −1 e−βHeff (q,p,Q) dqdp,
(32)
where
2 #
p2 1 2
λ
Heff (q, p, Q) =
+ ω q − 2Q
.
2
2
ω
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"
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This assumption implies that
q(0) =
p
λ
Q0 + β −1 ω −2 ξ,
2
ω
p(0) =
p
β −1 η,
(34)
where the ξ, η are independent N (0, 1) random variables.
This assumption ensures that the forcing term in (30) is mean
zero.
The fluctuation-dissipation theorem takes the form
hF (t)F (s)i = β −1 γ(t − s).
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We can perform the same calculation for the coupled particle-field
model (20). We obtain
Z t
0
Q̈ = −V (Q) −
D(t − s)Q(s) ds + hφ0 , e−Lt αi,
(36)
0
where
A=
0 1
∂x2 0
,
R
hf , hi = (∂x f1 ∂x h1 + f2h2 ) dx, f = (f1 , f2 ) and
α̂(k ) = −ik ρ̂(k )/k 2 , 0 in Fourier space.
Furthermore
D(t) = heLt Lα, αi = Ċ(t).
C(t) is the covariance of the Gaussian noise process
F (t) = hφ0 , e−Lt αi.
In particular:
E(F (t)F (s)) = β −1 C(t − s) = β −1
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Z
|ρ̂(k )|2 eikt dk .
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The spectral density of the autocorrelation function of the noise
process is the square of the Fourier transform of the density ρ(x)
which controls the coupling between the particle and the
environment.
The GLE (36) is equivalent to the original infinite dimensional
Hamiltonian system with random initial conditions.
Proving ergodicity, convergence to equilibrium etc for (36) is
equivalent to proving ergodicity and convergence to equilibrium for
the infinite dimensional Hamiltonian system.
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For general coupling functions ρ(x) the GLE (36) describes a
non-Markovian system. It can be represented as a Markovian
system only if we add an infinite number of additional variables.
However, for appropriate choices of the coupling function ρ(x) the
GLE (36) is equivalent to a Markovian process in a finite
dimensional extended phase space.
Definition
We will say that a stochastic process Xt is quasi-Markovian if it can be
represented as a Markovian stochastic process by adding a finite
number of additional variables: There exists a stochastic process Yt so
that {Xt , Yt } is a Markov process.
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Proposition
P
m
Assume that p(k ) = M
m=1 cm (−ik ) is a polynomial with real
coefficients and roots in the upper half plane then the Gaussian
process with spectral density |p(k )|−2 is the solution of the linear SDE
dWt
d
xt =
.
(37)
p −i
dt
dt
Proof.
The solution of (37) is
Z
t
k (t − s) dW (s),
xt =
−∞
1
k (t) = √
2π
Z
eikt
1
dk .
p(k )
We compute
Z
E(xt xs ) =
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eik (t−s)
1
dk .
|p(k )|2
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Example
Take p(k ) ∼ (ik + α). The spectral density is
|ρ̂(k )|2 =
α
1
.
2
π π + α2
The autocorrelation function is
Z
α
1
C(t) =
eikt 2
dk = e−|α|t .
π
k + α2
The linear SDE is
dxt = −αxt dt +
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√
2αdWt .
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Consider the GLE
0
2
Z
Q̈ = −V (Q) − λ
t
γ(t − s)Q̇(s) ds + λF (t),
(38)
0
with hF (t)F (s)i = β −1 γ(t − s) = β −1 e−α(t−s) .
F (t) is the stationary Ornstein-Uhlenbeck process:
p
dF
dW
= −αF + 2β −1 α
,
dt
dt
with F (0) ∼ N (0, β −1 ).
(39)
We can rewrite (38) as a system of SDEs:
dQ
dt
dP
dt
dZ
dt
= P,
= −V 0 (Q) + λZ ,
p
dW
= −αZ − λP + 2αβ −1
,
dt
where Z (0) ∼ N (0, β −1 ).
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The process {Q(t), P(t), Z (t)} ∈ R3 is Markovian.
It is a degenerate Markov process: noise acts directly only on one
of the 3 degrees of freedom.
The generator of this process is
L = p∂q + − V 0 (q) + λz ∂p − αz + λp ∂z + αβ −1 ∂z2 .
It is a hypoelliptic and hypocoercive operator.
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Rescale λ → λ , α →
α
2
Eqn. (40) becomes
dq = p dt,
dp = −V 0 (q ) dt +
dz = −
(41a)
λ z dt,
r
α λ
z dt − p dt +
2
(41b)
2αβ −1
dW ,
2
(41c)
In the limit as → 0 we obtain the Langevin equation for
q (t), p (t).
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Proposition
Let {q (t), p (t), z (t)} on R3 be the solution of (41) with
V (q) ∈ C ∞ (T) with stationary initial conditions. Then {q (t), p (t)}
converge weakly to the solution of the Langevin equation
p
dq = p dt, dp = −V 0 (q) dt − γp dt + 2γβ −1 dW ,
(42)
where the friction coefficient is given by the formula
γ=
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λ2
.
α
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(43)
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We have derived the (generalized) Langevin equation from a
”particle + field” model where the field is at equilibrium at t = 0.
The model that we can considered is very specific since the field
and the coupling are linear.
The GLE can be derived (at least formally) for more general
Hamiltonian systems using the Mori-Zwanzig formalism
(projection operator techniques for the Liouville equation).
The Markovian approximation of the GLE leads to a finite
dimensional hypoelliptic diffusion.
Conjecture (F. Nier): Assume that
L = −∇V · ∇ + β −1 ∆
has compact resolvent. Then so does
L == p · ∇q − ∇q V · ∇p + γ(−p · ∇p + β −1 ∆p ).
Formulate a similar conjecture for all Markovian approximations of
the GLE.
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LINEAR RESPONSE THEORY
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Let Xt denote a stationary dynamical system with state space X
and invariant measure µ(dx) = f∞ (x) dx.
We probe the system by adding a time dependent forcing F (t)
with 1 at time t0 .
Goal: calculate the distribution function f (x, t) of the perturbed
systems Xt , 1, in particular in the long time limit t → +∞.
We can then calculate expectation value of observables as well as
correlation functions.
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We assume that the distribution function f (x, t) satisfies a linear
kinetic equation e.g. the Liouville or the Fokker-Planck equation:
∂f ∂t
= L∗ f ,
f t=t = f∞ .
0
(44a)
(44b)
The choice of the initial conditions reflects the fact that at t = t0
the system is at equilibrium.
Since f∞ is the unique equilibrium distribution, we have that
L∗0 f∞ = 0.
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The operator L∗ can be written in the form
L∗ = L∗0 + L∗1 ,
(46)
where cL∗0 denotes the Liouville or Fokker-Planck operator of the
unperturbed system and L∗1 is related to the external forcing.
We will assume that L1 is of the form
L∗1 = F (t) · D,
(47)
where D is some linear (differential) operator.
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Example
(A deterministic dynamical system). Let Xt be the solution of the
differential equation
dXt
= h(Xt ),
(48)
dt
on a (possibly compact) state space X. We add a weak time
dependent forcing to obtain the dynamics
dXt
= h(Xt ) + F (t).
dt
(49)
We assume that the unperturbed dynamics has a unique invariant
distribution f∞ which is the solution of the stationary Liouville equation
∇ · h(x)f∞ = 0,
(50)
equipped with appropriate boundary conditions.
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Example
The operator L∗ in (46) has the form
L∗ · = −∇ · h(x) · − F (t) · ∇ · .
In this example, the operator D in (47) is D = −∇.
A particular case of a deterministic dynamical system of the
form (48), and the most important in statistical mechanics, is that
of an N-body Hamiltonian system.
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Example
(A stochastic dynamical system). Let Xt be the solution of the
stochastic differential equation
dXt = h(Xt ) dt + σ(Xt ) dWt ,
(51)
on Rd , where σ(x) is a positive semidefinite matrix and where the Itô
interpretation is used. We add a weak time dependent forcing to obtain
the dynamics
dXt = h(Xt ) dt + F (t) dt + σ(Xt ) dWt .
(52)
We assume that the unperturbed dynamics has a unique invariant
distribution f∞ which is the solution of the stationary Fokker-Planck
equation
1
(53)
− ∇ · h(x)f∞ + D 2 : Σ(x)f∞ = 0,
2
where Σ(x) = σ(x)σ T (x).
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Example
The operator L∗ in (46) has the form
1
L∗ · = −∇ · h(x) · + D 2 : Σ(x) · − F (t) · ∇.̇
2
As in the previous example, the operator D in (47) is D = −∇.
A particular case of Example 5 is the Langevin equation:
p
q̈ = −∇V (q) + F (t) − γ q̇ + 2γβ Ẇ .
(54)
We can also add a forcing term to the noise in (51), i.e. we can
consider a time-dependent temperature. For the Langevin
equation the perturbed dynamics is
q
q̈ = −∇V (q) + F (t) − γ q̇ + 2γβ −1 (1 + T (t))Ẇ ,
(55)
with 1 + T (t) > 0. in this case the operator L∗1 is
L∗1 = −F (t) · ∇p + γβ −1 T (t)∆p ,
where p = q̇.
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We proceed with the analysis of (44). We look for a solution in the
form of a power series expansion in :
f = f0 + f1 + . . . .
We substitute this into (44a) and use the initial condition (44b) to
obtain the equations
∂f0
= L∗0 f0 ,
∂t
f0 t=0 = f∞ ,
∂f1
= L∗0 f1 + L∗1 f0 ,
∂t
The only solution to (56a) is
f1 t=0 = 0.
(56a)
(56b)
f0 = f∞ .
We use this into (56b) and use (47) to obtain
∂f1
= L∗0 f1 + F (t) · Df∞ ,
∂t
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f1 t=0 = 0.
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We use the variation of constants formula to solve this equation:
Z
t
f1 (t) =
∗
eL0 (t−s) F (s) · Df∞ ds.
(57)
t0
Now we can calculate the deviation in the expectation value of an
observable due to the external forcing: Let h·ieq and h·i denote the
expectation value with respect to f∞ and f , respectively.
Let A(·) be an observable and denote by A(t) the deviation of its
expectation value from equilibrium, to leading order:
A(t) := hA(Xt )i − hA(Xt )ieq
Z
=
A(x) f (x, t) − feq (x) dx
Z t
Z
L∗0 (t−s)
= A(x)
e
F (s) · Df∞ ds dx.
t0
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
42 / 67
Assuming now that we can interchange the order of integration we
can rewrite the above formula as
Z t
Z
L∗0 (t−s)
A(t) = A(x)
e
F (s) · Df∞ ds dx
t0
Z t Z
=
A(x)e
L∗0 (t−s)
· Df∞ dx
ds
t0
Z t
=: t0
RL0 ,A (t − s)F (s) ds,
where we have defined the response function
Z
∗
RL0 ,A (t) = A(x)eL0 t · Df∞ dx
(58)
(59)
We set now the lower limit of integration in (58) to be t0 = −∞
(define RL0 ,A (t) in (59) to be 0 for t < 0) and assume that we can
extend the upper limit of integration to +∞ to write
Z +∞
A(t) = RL0 ,A (t − s)F (s) ds.
(60)
−∞
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
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As expected (since we have used linear perturbation theory), the
deviation of the expectation value of an observable from its
equilibrium value is a linear function of the forcing term.
(60) has the form of the solution of a linear differential equation
with RL0 ,A (t) playing the role of the Green’s function. If we
consider a delta-like forcing at t = 0, F (t) = δ(t), then the above
formula gives
A(t) = RL0 ,A (t).
Thus, the response function gives the deviation of the expectation
value of an observable from equilibrium for a delta-like force.
Consider a constant force that is exerted to the system at time
t = 0, F (t) = F Θ(t) where Θ(t) denotes the Heaviside step
function. For this forcing (58) becomes
Z
A(t) = F
0
G.A. Pavliotis (MICMAC-IC)
t
RL0 ,A (t − s) ds.
Open Classical Systems
(61)
44 / 67
There is a close relation between the response function (59) and
stationary autocorrelation functions.
Let Xt be a stationary Markov process in Rd with generator L and
invariant distribution f∞ .
Let A(·) and B(·) be two observables.
The stationary autocorrelation function hA(Xt )B(X0 )ieq can be
calculated as follows
κA,B (t) := hA(Xt )B(X0 )ieq
Z Z
=
A(x)B(x0 )p(x, t|x0 , 0)f∞ (x0 ) dxdx0
Z Z
∗
=
A(x)B(x0 )eL t δ(x − x0 )f∞ (x0 ) dxdx0
Z Z
=
eLt A(x)B(x0 )δ(x − x0 )f∞ (x0 ) dxdx0
Z
=
eLt A(x)B(x)f∞ (x) dx,
where L acts on functions of x.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
45 / 67
Thus we have established the formula
κA,B (t) = St A(x), B(x) f∞ ,
(62)
where St denotes the semigroup generated by L and ·, · f∞
denotes the L2 inner product weighted by the invariant distribution
of the diffusion process.
−1 Df . We
Consider now the particular choice B(x) = f∞
∞
combine (59) and (62) to deduce
κA,f∞
(t) = RL0 ,A (t).
−1
Df∞
(63)
This is a version of the fluctuation-dissipation theorem.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
46 / 67
Example
Consider the Langevin dynamics with an external forcing F .
p
dq = p dt, dp = −V 0 (q) dt + F dt − γp dt + 2γβ −1 dWt .
We have D = −∂p and
−1
B = f∞
Df∞ = βp.
We use (63) with A = p:
βhp(t)p(0)ieq = RL0 ,p (t).
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
47 / 67
Example (continued)
When the potential is harmonic, V (q) = 12 ω02 q 2 , we can compute
explicitly the response function and, consequently, the velocity
autocorrelation function at equilibrium:
r
1 − γt
γ2
e 2 sin(ω1 t), ω1 = ω02 −
Rq,L (t) =
ω1
4
and
γt
Rp,L (t) = e− 2
γ
cos(ω1 t) −
sin(ω1 t) .
2ω1
Consequently:
γt
hp(t)p(0)ieq = β −1 e− 2
cos(ω1 t) −
γ
sin(ω1 t) .
2ω1
Similar calculations can be done for all linear SDEs.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
48 / 67
Example
Consider the Langevin dynamics with a perturbation in the
temperature
q
dq = p dt, dp = −V 0 (q) dt + F dt − γp dt + 2γβ −1 (1 + F ) dWt .
We have D = γβ −1 ∂p2 and
−1
B = f∞
Df∞ = γβ(p2 − β −1 ).
Let H(p, q) = p2 /2 + V (q) denote the total energy. We have
−1 ∗
f∞
L0 H(p, q)f∞ = L0 H(p, q) = γ(−p2 + β −1 ).
d
Consequently (using (62)): κA,f∞
(t) = −β dt
κA,H (t).
−1
Df∞
d
For A = H we obtain RH,L (t) = −β dt
hH(t)H(0)ieq .
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
49 / 67
We calculate the long time limit of A(t) when the external forcing is
a step function.
The following formal calculations can be justified e.g. for reversible
diffusions using functional calculus.
Z
t
Z tZ
RL,A (t − s) ds =
0
A(x)eL0 (t−s) Df∞ dx ds
0
Z Z t
eL(t−s) A(x) Df∞ ds dx
0
Z Z t
L0 t
L(−s)
=
e
e
dsA(x) Df∞ dx
0
Z eL0 t (−L0 )−1 eL0 (−t) − I A(x) Df∞ dx
=
Z =
I − eL0 t (−L)−1 A(x) Df∞ dx
=
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
50 / 67
Assuming now that limt→+∞ eLt = 0 (again, think of reversible
diffusions) we have that
Z
D := lim
t→+∞ 0
t
Z
RL,A (t − s) ds =
(−L)−1 A(x)Df∞ dx.
Using this in (61) and relabeling F 7→ F we obtain
Z
A(t)
= (−L)−1 A(x)Df∞ dx.
lim lim
F →0 t→+∞ F
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
(64)
51 / 67
Remark
At least formally, we can interchange the order with which we take
the limits in (64).
Formulas of the form (64) enable us to calculate transport
coefficients, such as the diffusion coefficient.
We can rewrite the above formula in the form
Z
A(t)
lim lim
= φDf∞ dx,
F →0 t→+∞ F
(65)
where φ is the solution of the Poisson equation (when A(x) is
mean zero)
− Lφ = A(x),
(66)
equipped with appropriate boundary conditions.
This is precisely the formalism that we obtain using
homogenization theory.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
52 / 67
Consider the Langevin dynamics in a periodic or random potential.
p
dqt = pt dt, dpt = −∇V (qt ) dt − γpt dt + 2γβ −1 dW .
From Einstein’s formula (10) we have that the diffusion coefficient
is related to the mobility according to
hpt i
F →0 t→+∞ F
D = β −1 lim lim
where we have used hpt ieq = 0.
We use now (65) with A(t) = pt , D = −∇p , f∞ = Z1 e−βH(q,p) to
obtain
Z Z
D=
φpf∞ dpdq = h−Lφ, φif∞ ,
(67)
which is the formula obtained from homogenization theory (e.g.
Kipnis and Varadhan 1985, Rodenhausen 1989).
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
53 / 67
Consider the unperturbed dynamics
dq = p dt,
dp = −V 0 (q) dt − γp dt +
p
2γβ −1 dW ,
(68)
where V is a smooth periodic potential. Then the rescaled process
q (t) := q(t/2 ),
Converges to a Brownian motion with diffusion coefficient
Z Z
D=
((−L)−1 p)pf∞ dpdq.
(69)
A similar result can be proved when V is a random potential.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
54 / 67
Consider now the perturbed dynamics
dqF = pF dt,
dpF = −V 0 (qF ) dt + F dt − γpF dt +
p
2γβ −1 dW .
(70)
At least for F sufficiently small, the dynamics is ergodic on T × R
F (p, q) which is a differentiable
with a smooth invariant density f∞
function of F .
The external forcing induces an effective drift
Z Z
F
VF =
pf∞
dpdq.
(71)
The mobility is defined as
µ :=
G.A. Pavliotis (MICMAC-IC)
d
VF .
dF
F =0
Open Classical Systems
(72)
55 / 67
Theorem (Rodenhausen J. Stat. Phys. 1989)
The mobility is well defined by (72) and
D = β −1 µ.
(73)
Proof.
Ergodic theory for hypoelliptic diffusions.
Study of the Poisson and stationary Fokker-Planck equations.
Girsanov’s formula.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
56 / 67
Upon combining (63) with (64) we obtain
Z
D = lim
t→+∞ 0
t
κA,f∞
(t − s) ds.
−1
Df∞
(74)
Thus, a transport coefficient can be computed in terms of the time
integral of an appropriate autocorrelation function.
This is an example of the Green-Kubo formula
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
57 / 67
We can obtain a more general form of the Green-Kubo formalism.
We define the generalized drift and diffusion coefficients as
follows :
1 V f (x) = lim E f (Xh ) − f (X0 )X0 = x = Lf
(75)
h→0 h
and
D f ,g (x) :=
=
1 E (f (Xt+h ) − f (Xt ))((g(Xt+h ) − g(Xt ))Xt = x
h→0 h
L(fg)(x) − (gLf )(x) − (f Lg)(x),
(76)
lim
where f , g are smooth functions (in fact, all we need is f , g ∈ D(L)
and fg ∈ D(L0 )).
The equality in (75) follows from the definition of the generator of a
diffusion process.
We will prove the equality in (75).
Sometimes D f ,g (x) is called the opérateur carré du champ .
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
58 / 67
To prove (76), notice first that, by stationarity, it is sufficient to
prove it at t = 0. Furthermore
(f (Xh ) − f (X0 )) (g(Xh ) − g(X0 )) = (fg)(Xh ) − (fg)(X0 )
−f (X0 )(g(Xh ) − g(X0 ))
−g(X0 )(f (Xh ) − f (X0 )).
Consequently:
D f ,g (x) :=
=
=
G.A. Pavliotis (MICMAC-IC)
1 E (f (Xh ) − f (X0 ))((g(Xh ) − g(X0 ))X0 = x
h→0 h
1 lim E (fg)(Xh ) − (fg)(X0 )X0 = x
h→0 h
1 − lim E g(Xh ) − g(X0 )X0 = x f (x)
h→0 h
1 − lim E f (Xh ) − f (X0 )X0 = x g(x)
h→0 h
L(fg)(x) − (gLf )(x) − (f Lg)(x).
lim
Open Classical Systems
59 / 67
We have the following result.
Theorem
(The Green-Kubo formula) Let Xt be a stationary reversible diffusion
process with state space X, generator L, invariant measure µ(dx) and
let V f (x), D f ,g (x). Then
Z
Z ∞ 1
f ,g
f
g
D µ(dx) =
E V (Xt )V (X0 ) dt.
(77)
2
0
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
60 / 67
Proof.
Let (·, ·)µ denote the inner product in L2 (X, µ). We note that
Z
1
D f ,g µ(dx) = (−Lf , g)µ ,
2
In view of (78), formula (77) becomes
Z ∞ (−Lf , g)µ =
E V f (Xt )V g (X0 ) dt.
(78)
(79)
0
R∞
Now we use (62), together with the identity 0 eLt · dt = (−L)−1
to obtain
Z ∞
κA,B (t) dt = ((−L)−1 A, B)µ .
0
We set now A = V f = Lf , B = V g = Lg in the above formula to
deduce (79) from which (77) follows.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
61 / 67
Remark
The above formal calculation can also be performed in the
nonreversible case to obtain
Z
Z ∞ 1
f ,f
E V f (Xt )V f (X0 ) dt.
D µ(dx) =
2
0
In the reversible case (i.e. L being a selfadjoint operator in
H := L2 (X, µ)) the formal calculations can be justified using
functional calculus.
For the reversible diffusion process
dXt = −∇V (Xt ) dt +
p
2β −1 dWt
and under appropriate assumptions on the potential, the
generator L has compact resolvent and its eigenfunctions form an
orthonormal basis in H. In this case the proof of (11) becomes
quite simple.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
62 / 67
The spectral representation of L is
Z 0
L=
λ dEλ ,
−∞
from which it follows that
Z 0
eLt =
eλt dEλ
and eLt L =
−∞
Z
0
eλt λ.
−∞
Remember that V f = Lf We calculate (with St = eLt )
Z 0
E[V f (Xt )V g (X0 )] = (St Lf , Lg)µ =
λ2 eλt d(Eλ f , g)µ .
(80)
−∞
From Fubini’s theorem and Cauchy-Schwarz it follows that
Z ∞Z 0
λ2 eλt |d(Eλ f , g)µ | dt ≤ DL (f )1/2 DL (g)1/2 < +∞
0
−∞
with DL (f ) := (−Lf , f )µ .
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
63 / 67
We use now (80) and Fubini’s theorem to calculate
Z
+∞
f
g
t
Z
E[V (Xt )V (X0 )] dt
(St Lf , Lg)µ dt
=
0
0
+∞ Z 0
Z
=
−∞
0
Z
λ2 eλt d(Eλ f , g)µ dt
0
(−λ)d(Eλ f , g)µ
=
−∞
Z
0
Z
(−λ)dEλ f ,
=
−∞
= (−Lf , g)µ =
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
dEλ g
−∞
1
2
Z
!
0
µ
D f ,g (x)µ(dx).
64 / 67
Example
Consider the diffusion process
dXt = b(Xt ) dt + σ(Xt ) ◦ dWt ,
(81)
where the noise is interpreted in the Stratonovich sense.
The generator is
1
L· = b(x) · ∇ + ∇ · (A(x)∇·) ,
2
(82)
where A(x) = (σσ T )(x).
We assume that the diffusion process has a unique invariant
distribution which is the solution of the stationary Fokker-Planck
equation
L∗ ρ = 0.
(83)
When A is strictly positive definite ergodicity follows from PDE
arguments.
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
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Example (Continued)
The stationary process Xt (i.e. X0 ∼ ρ(x)dx) is reversible provided
that
1
b(x) = A(x)∇ log ρ(x).
(84)
2
Let f = xi , g = xj . We calculate
1
V xi (x) = Lxi = bi + ∂k Aik , i = 1, . . . d.
2
(85)
We use the detailed balance condition (84) and (78) to calculate
Z
Z 1
1
xi ,xj
bi (x) + ∂k Aik (x) xj ρ(x) dx
D µ(dx) = = −
2
2
Z
1
= −
(Aik ∂k ρ(x) + ∂k Aik (x)ρ(x)) xj dx
2
Z
1
=
Aij (x)ρ(x) dx.
2
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
66 / 67
Example (Continued)
The Green-Kubo formula (77) gives:
Z
Z +∞ 1
Aij (x)ρ(x) dx =
E V xi (Xt )V xj (X0 ) dt,
2
0
(86)
where the drift V xi (x) is given by (84).
G.A. Pavliotis (MICMAC-IC)
Open Classical Systems
67 / 67