Nonequilibrium diusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator

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Nonequilibrium diﬀusion processes via non-Hermitian electromagnetic quantum

mechanics

with application to the statistics of entropy production in the Brownian gyrator

Alain Mazzolo

Universit´e Paris-Saclay, CEA, Service d’ ´

Etudes des R´eacteurs et de Math´ematiques Appliqu´ees, 91191, Gif-sur-Yvette, France

C´ecile Monthus

Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique, 91191 Gif-sur-Yvette, France

The nonequilibrium Fokker-Planck dynamics in an arbitrary force ﬁeld ~

f(~x) in dimension Nis

revisited via the correspondence with the non-Hermitian quantum mechanics in a real scalar poten-

tial V(~x) and in a purely imaginary vector potential [−i~

A(~x)] of real amplitude ~

A(~x). The relevant

parameters of irreversibility are then the N(N−1)

2magnetic matrix elements Bnm (~x) = −Bmn(~x) =

∂nAm(~x)−∂mAn(~x), while it is enlightening to explore the corresponding gauge transformations of

the vector potential ~

A(~x). This quantum interpretation is even more fruitful to study the statistics

of all the time-additive observables of the stochastic trajectories, since their generating functions

correspond to the same quantum problem with additional scalar and/or vector potentials. Our main

conclusion is that the analysis of their large deviations properties and the construction of the corre-

sponding Doob conditioned processes can be drastically simpliﬁed via the choice of an appropriate

gauge for each purpose. This general framework is then applied to the special time-additive observ-

ables of Ornstein-Uhlenbeck trajectories in dimension N, whose generating functions correspond to

quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e. to quan-

tum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we ﬁnally

focus on the Brownian gyrator in dimension N= 2 to compute the large deviations properties of

the entropy production of its stochastic trajectories and to construct the corresponding conditioned

processes having a given value of the entropy production per unit time.

I. INTRODUCTION

A. On the various links between diﬀusion processes and quantum mechanics

1. Link between the Brownian motion and the Euclidean quantum mechanics for a free particle

For the Brownian motion in dimension N, the probability P(~x, t) to be at position ~x at time tsatisﬁes the heat

equation

∂tP(~x, t) = 1

2∆P(~x, t)≡1

∇2P(~x, t)≡1

n=1

∂2

nP(~x, t) (1)

which involves the Laplacian ∆ = ~

∇2built from the spatial derivatives ∂n≡∂

∂xnwith respect to the Ncoordinates

xnfor n= 1,2, ., N. The heat Eq. 1 corresponds to the Euclidean-time t=iθ version of the quantum mechanics for

a free particle, where the amplitude ψ(~x, θ) to be at position ~x at time θsatisﬁes the free Schr¨odinger equation that

involves only the Laplacian

i∂θψ(~x, θ) = −1

2∆ψ(~x, θ) (2)

This correspondence at the level of generators is of course even more powerful at the level of Feynman path-integrals

for trajectories [1]. It is thus very natural to extend this analogy as much as possible by considering the Euclidean-time

quantum mechanics for a particle in an electromagnetic potential (see the reminder in Appendix A), with the various

special cases recalled in the next subsections.

2. Similarity transformation between detailed-balance diﬀusions and supersymmetric quantum mechanics

As described in textbooks [2–5], the generator of a Markov processes satisfying detailed-balance can be transformed

via a similarity transformation into an Hermitian operator, with the very important spectral consequences. For the

arXiv:2210.05353v2 [cond-mat.stat-mech] 4 Jan 2023

simplest example of a diﬀusion process of diﬀusion coeﬃcient D= 1/2 converging toward the normalizable steady

state P∗(~x) that can always be rewritten in terms of some function φ(~x) in the exponential

P∗(~x) = P∗(~

0)e−2φ(~x)(3)

the corresponding Fokker-Planck dynamics

∂tP(~x, t) = −~

∇.P(~x, t)~

frev (~x)−1

∇P(~x, t)(4)

satisfying detailed-balance involves the force

frev (~x) = 1

∇ln P∗(~x) = −~

∇φ(~x) (5)

The similarity transformation

P(~x, t) = pP∗(~x)ψ(~x, t) = qP∗(~

0) e−φ(~x)ψ(~x, t) (6)

transforms the Fokker-Planck Eq. 4 into the Euclidean Schr¨odinger equation for ψ(~x, t)

∂tψ(~x, t) = −Hψ(~x, t) (7)

which involves the well-known Hermitian quantum supersymmetric Hamiltonian H(see the review [6] and references

therein)

H=H†=1

2−~

∇+ (~

∇φ(~x)).~

∇+ (~

∇φ(~x))≡ −1

2∆ + V(~x) (8)

with the very speciﬁc form of the scalar potential

V(~x) = 1

2[~

∇φ(~x)]2−∆φ(~x)(9)

while the quantum-normalized zero-energy ground-state reads

ψGS (~x) = pP∗(~x) = qP∗(~

0) e−φ(~x)(10)

3. Feynman-Kac formula to analyze the time-additive observables of the stochastic trajectories

The standard method to study the statistics of the time-additive observables of Markov trajectories is the introduc-

tion of the appropriate deformations of the Markov generator. This approach goes back to the famous Feynman-Kac

formula [1, 7–10] introduced to analyze any observable O[~x(0 ≤s≤t)] of the Brownian trajectory ~x(0 ≤τ≤t) that

can be parametrized by some scalar ﬁeld V[O](~x) and by some vector ﬁeld ~

A[O](~x) in the Stratonovich interpretation

O[~x(0 ≤τ≤t)] = Zt

dτ h−V[O](~x(τ)) + ˙

~x(τ).~

A[O](~x(τ))i(11)

Its generating function Z[k](~x, t|~y, 0) of parameter kover the Brownian trajectories ~x(0 ≤s≤t) starting at ~x(0) = ~y

and ending at ~x(t) = ~x can be written as the Feynman path-integral

Z[k](~x, t|~y, 0)≡δ(N)(~x(t)−~x)ekO[~x(0≤τ≤t)]δ(N)(~x(0) −~y)

=Z~x(τ=t)=~x

~x(τ=0)=~y D~x(τ)e

−Zt

dτ ˙

~x2(τ)

2+kZt

dτ h−V[O](~x(τ)) + ˙

~x(τ).~

A[O](~x(τ))i

≡ h~x|e−tH[k]|~yi(12)

that corresponds to the Euclidean quantum propagator h~x|e−tH[k]|~xiassociated to the Hamiltonian

H[k]=−1

2~

∇ − k~

A[O](~x)2+kV [O](~x) = 1

2−i~

∇+ik ~

A[O](~x)2+kV [O](~x) (13)

Since one is usually interested into real observables O[~x(0 ≤τ≤t)] parametrized by real ﬁelds V[O](~x) and ~

A[O](~x),

it is useful to distinguish the three following cases :

(i) If ~

A[O](~x)≡0 in the additive observable of Eq. 11, then the Hamiltonian of Eq. 13 is Hermitian

H[k]= (H[k])†=−1

2∆ + kV [O](~x) (14)

and involves only the scalar potential (kV [O](~x)).

(ii) If V[O](~x)≡0 in the additive observable of Eq. 11, then it is possible to consider instead k=iq with real qto

transform the generating function of Eq. 12 into the characteristic function of the observable O[~x(0 ≤τ≤t)] for the

Fourier parameter q. Then the Hamiltonian of Eq. 13 is complex but Hermitian

H[k=iq]= (H[k=iq])†=−1

2~

∇ − iq ~

A[O](~x)2=1

2−i~

∇ − q~

A[O](~x)2(15)

and involves only the real vector potential (qA[O](~x)) from the quantum point of view. Let us mention that while the

Feynman-Kac formula is often described only for the scalar potential case of Eq. 14, its application for the vector

potential case of Eq. 15 plays a major role to take into account topological constraints in the context of polymer

physics [11, 12] and to analyze the winding properties of Brownian paths [8, 13–18].

(iii) For the general case of an additive observable of Eq. 11 where both ﬁelds V[O](~x)6= 0 and ~

A[O](~x)6= 0 are

nonvanishing, it is clear that the Hamiltonian of Eq. 13 is diﬀerent from its adjoint

H[k]6= (H[k])†=−1

2~

∇+k~

A[O](~x)2+kV [O](~x) = 1

2−i~

∇ − ik ~

A[O](~x)2+kV [O](~x) (16)

and corresponds to the quantum problem with the real scalar potential kV [O](~x) and the purely imaginary vector

potential (−ikA[O](~x)), described in Appendix A around Eqs A19 and A20.

It is also important to stress that in the ﬁeld of large deviations discussed in the next subsection, even for the

case (ii) discussed above, it is standard to consider only the generating function Z[k](~x, t|~y, 0) for real k, and not

the characteristic function corresponding to k=iq with real q, i.e. it is usual to work with the real non-Hermitian

Hamiltonian H[k]6= (H[k])†instead of the complex Hermitian Hamiltonian H[k=iq]= (H[k=iq])†of Eq. 15.

B. Large deviations properties for trajectory observables of Markov trajectories

The theory of large deviations (see the reviews [19–21] and references therein) has become the unifying language in

the ﬁeld of nonequilibrium processes (see the reviews with diﬀerent scopes [22–30], the PhD Theses [31–36] and the

Habilitation Thesis [37]). In particular, the approach based on the deformed Markov generators recalled above has

been used to analyze the statistics of many interesting additive observables of various Markov processes over the years

[22, 27–30, 32, 37–78]. While the large deviations properties of all types of Markov processes are of course interesting,

the following summary is restricted to the case of Markov processes converging towards a steady state.

1. Rate functions I(o)and scaled-cumulant-generating-functions E(k)of time-additive observables

Since a time-additive observable O[~x(0 ≤s≤t)] of a Markov trajectory ~x(0 ≤s≤t) is extensive with respect to

the duration t, it is useful to introduce its rescaled intensive counterpart

o[~x(0 ≤s≤t)] ≡O[~x(0 ≤s≤t)]

t→+∞o∗(17)

which will converge for t→+∞towards the steady value o∗that can be computed from the steady state properties.

For large t, the ﬂuctuations around this steady-state value o∗are described by the following large deviations form for

the probability Pt(o) to see the intensive value oover the time-window t

Pt(o)'

t→+∞e−tI(o)(18)

The positive rate function I(o)≥0 vanishes only for the steady value o∗where it is minimum

0 = I(o∗) = I0(o∗) (19)

For large time t→+∞, the generating function Z[k](~x, t|~y, 0) rewritten as Euclidean Schr¨odinger propagator

associated to some Hamiltonian H[k](as in the example of Eq. 12 for Brownian trajectories) will display the asymptotic

behavior

Z[k](~x, t|~y, 0) = h~x|e−tH[k]|~yi '

t→+∞e−tE(k)rk(~x)lk(~y) (20)

where E(k) is the ground-state energy of the Hamiltonian H[k], while rk(.) and lk(.) are the corresponding positive

right and left eigenvectors

E(k)rk(~x)= H[k]rk(~x)

E(k)lk(~x)= (H[k])†lk(~x) (21)

with the normalization

ZdN~x lk(~x)rk(~x) = 1 (22)

For k= 0 where the generating function reduces to the propagator P(~x, t|~y, 0) of the Markov process that converges

towards the steady state P∗(x) for any initial condition y

Z[k=0](~x, t|~y, 0) = P(~x, t|~y, 0) '

t→+∞e−tE(0)r0(~x)l0(~y) = P∗(x) (23)

the ground state energy vanishes E(k= 0) = 0, while the right eigenvector corresponds to the steady state rk=0(~x) =

P∗(~x) and the left eigenvector is trivial lk=0(~y) = 1

E(0)= 0

r0(~x)= P∗(x)

l0(~y)= 1 (24)

The consistency between the asymptotic time behavior of Eq. 20 and the large deviation form of Eq. 18 via the

saddle-point evaluation for large t

hekO[~x(0≤s≤t)]i=hekto[~x(0≤s≤t)]i ≡ ZdoektoPt(o)'

t→+∞Zdoet[ko−I(o)] '

t→+∞e−tE(k)(25)

yields that the ground-state energy E(k) is the scaled-cumulant-generating-function and corresponds the Legendre

transform of the rate function I(o)

ko −I(o)= −E(k)

k−I0(o)= 0 (26)

So the reciprocal Legendre transform

ko +E(k)= I(o)

o+E0(k)= 0 (27)

allows us to compute the rate function I(o) from the knowledge of the energy E(k). In particular, the steady value

o∗satisfying Eq. 19 is conjugated to the value k= 0 and thus corresponds to the ﬁrst-order perturbation theory in k

of the ground-state energy E(k) around E(k= 0) = 0 in Eq. 27

o∗=−E0(k= 0) (28)

Once one has elucidated the large deviations properties of the observable via its rate function I(o), it is often interesting

to analyze the rare Markov trajectories that have been able to produce a given anomalous value o6=o∗diﬀerent from

the steady value o∗via the notion of canonical conditioning recalled in the next subsection.

2. Canonical conditioning of parameter kbased on the generating function Z[k](~x, t|~y, 0)

As explained in detail in the two complementary papers [55, 56] and in the Habilitation thesis [37], the canonical

conditioning of parameter kbased the generating function Z[k](~x, t|~y, 0) that is summarized below becomes equivalent

in the large-time limit t→+∞with the microcanonical conditioning that would impose the Legendre value o=

−E0(k) of Eq. 27 for the intensive observable o. The idea is that for each value k, one introduces the conditional

probability PCond[k](~z, τ) to be at position ~z at the internal time τ∈]0, t[

PCond[k](~z, τ ) = Z[k](~x, t|~z, τ)Z[k](~z, τ |~y, 0)

Z[k](~x, t|~y, 0) (29)

which is normalized over ~z at any time τ

ZdN~z PCond[k](~z, τ) = 1 (30)

and that satisﬁes the boundary conditions at times τ= 0 and τ=t

PCond[k](~z, τ = 0)= δ(N)(~z −~y)

PCond[k](~z, τ =t)= δ(N)(~z −~x) (31)

For large time t→+∞, the conditional probability PCond[k](~z, τ) of Eq. 29 at any interior time τsatisfying

0τtcan be evaluated from the asymptotic property of Eq. 20 for the three involved generating functions

PCond[k](~z, τ )'

0τt

e−(t−τ)E(k)rk(~x)lk(~z)e−τ E(k)rk(~z)lk(~y)

e−tE(k)rk(~x)lk(~y)

=lk(~z)rk(~z)≡ PCond[k]Interior (~z) (32)

to obtain that it does not depend on the interior time τand that it reduces to the product of the left eigenvector

lk(~z) and the right eigenvector rk(~z) of Eqs 21 and 22. The knowledge of the left eigenvector lk(~z) is then necessary

to construct the generator of the conditioned process that has Eq. 32 as steady state (see subsection III D of the main

text for more details).

This canonical conditioning of parameter kin the large-time limit t→+∞is thus a huge simpliﬁcation with

respect to the ﬁnite-time Doob process conditioned to end at the given position x(t) = xand at the given value

O[~x(0 ≤s≤t)] = Oof the additive observable, whose construction requires the knowledge of the ﬁnite-time joint

propagator P(~x, O, t|~y, 0,0) and produces time-dependent generators, as described for the various examples studied

recently [79–83].

C. Goals of the present paper

In the present paper, the main goal is to analyze the statistics of time-additive observables of Eq. 11 when

the diﬀusion process ~x(t) satisﬁes the Langevin stochastic diﬀerential system involving the Nindependent Wiener

processes wn(t)

dxn(t)= fn(~x(t)) dt +dwn(t) (33)

where the space-dependent force ~

f(~x) ensures that the corresponding Fokker-Planck dynamics for the propagator

P(~x, t|~y, 0) [i.e. the probability distribution to be at ~x at time twhen starting at ~y at time t= 0]

∂tP(~x, t|~y, 0) = −

n=1

∂nfn(~x)P(~x, t|~y, 0) −1

2∂nP(~x, t|~y, 0)≡ −HP(~x, t|~y, 0) (34)

converges towards some normalized steady state P∗(~x). The interpretation of the Fokker-Planck Eq. 34 as an

Euclidean Schr¨odinger equation involves the non-Hermitian quantum Hamiltonian H 6=H†

H=~

∇.−1

∇+~

f(~x)=−1

∇2+~

f(~x).~

∇+ [~

∇.~

f(~x)]

H†=−1

∇+~

f(~x).~

∇=−1

∇2−~

f(~x).~

∇(35)

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Nonequilibriumdiusionprocessesvianon-HermitianelectromagneticquantummechanicswithapplicationtothestatisticsofentropyproductionintheBrowniangyratorAlainMazzoloUniversiteParis-Saclay,CEA,Serviced'EtudesdesReacteursetdeMathematiquesAppliquees,91191,Gif-sur-Yvette,FranceCecileMonthusUniversitePa...

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Nonequilibrium diusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator

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