The role of multiplicative noise in critical dynamics

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Nathan O. Silvanoa,b,∗, Daniel G. Barcib

aCenter for Advanced Systems Understanding, Untermarkt 20, 02826 G¨orlitz, Helmholtz-Zentrum Dresden-Rossendorf,

Bautzner Landstraße 400, 01328 Dresden

bDepartamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524, 20550-013,

Rio de Janeiro, RJ, Brazil

Abstract

We study the role of multiplicative stochastic processes in the description of the dynamics of an order pa-

rameter near a critical point. We study equilibrium as well as out-of-equilibrium properties. By means of a

functional formalism, we build the Dynamical Renormalization Group equations for a real scalar order pa-

rameter with Z2symmetry, driven by a class of multiplicative stochastic processes with the same symmetry.

We compute the ﬂux diagram using a controlled ϵ-expansion, up to order ϵ2. We ﬁnd that, for dimensions

d= 4−ϵ, the additive dynamic ﬁxed point is unstable. The ﬂux runs to a multiplicative ﬁxed point driven by

a diﬀusion function G(ϕ) = 1 + g∗ϕ2(x)/2, where ϕis the order parameter and g∗=ϵ2/18 is the ﬁxed point

value of the multiplicative noise coupling constant. We show that, even though the position of the ﬁxed

point depends on the stochastic prescription, the critical exponents do not. Therefore, diﬀerent dynamics

driven by diﬀerent stochastic prescriptions (such as Itˆo, Stratonovich, anti-Itˆo and so on) are in the same

universality class.

Keywords: Classical phase transitions, Dynamical Renormalization Group, Stochastic dynamics

Contents

1 Introduction 3

2 Brief review of multiplicative stochastic processes 4

2.1 Singlestochasticvariable ...................................... 4

2.2 Multiplestochasticvariables..................................... 6

2.3 Continuumsystems.......................................... 7

2.4 Equilibriumproperties........................................ 8

3 Functional formalism 9

3.1 Linear response and the Fluctuation Dissipation Theorem . . . . . . . . . . . . . . . . . . . . 10

3.2 Real scalar ﬁeld with Z2symmetry................................. 11

4 Dynamical Renormalization Group 12

4.1 DRGTransformation ........................................ 12

4.2 ϵ-expansion .............................................. 14

4.2.1 TheGaussianFixedPoint.................................. 18

4.2.2 The non-Gaussian Multiplicative Fixed Point . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Corrections to scaling in multiplicative systems . . . . . . . . . . . . . . . . . . . . . . . . . . 22

∗Corresponding author

Email addresses: nathanosilvano@gmail.com (Nathan O. Silvano), daniel.barci@gmail.com (Daniel G. Barci)

Preprint submitted to Elsevier September 26, 2023

arXiv:2210.11969v2 [cond-mat.stat-mech] 23 Sep 2023

5 Summary and discussions 23

Appendix A DRG-equations 24

Appendix B Eﬀect of higher order multiplicative couplings 25

Appendix C Diagrammatic computations 27

Appendix D Rescaling of the action 31

1. Introduction

Multiplicative stochastic processes have a long history and they were used to model very diﬀerent dynam-

ical systems [1, 2]. Perhaps, the simplest example is the diﬀusion of a Brownian particle near a wall [3, 4].

Some other typical examples of multiplicative noise processes are micromagnetic dynamics [5, 6, 7] and

non-equilibrium transitions into absorbing states [8]. Moreover, multiplicative noise plays a central role in

the description of out-of equilibrium systems [9, 10] and noise-induced phase transitions [11, 12].

In this paper, we study the role of multiplicative noise in the dynamics of phase transitions. Out-of-

equilibrium evolution near continuous phase transitions is a fascinating subject. While equilibrium properties

are strongly constrained by symmetry and dimensionality, the dynamics is much more involved and it

generally depends on conserved quantities and other details of the system. The interest in critical dynamics

is rapidly growing up in part due to the wide range of multidisciplinary applications in which criticality has

deeply impacted. For instance, the collective behavior of diﬀerent biological systems has critical properties,

displaying space-time correlation functions with nontrivial scaling laws [13, 14]. Other interesting examples

come from epidemic spreading models where dynamic percolation is observed near multicritical points [15].

Moreover, strongly correlated systems, such as antiferromagnets in transition-metal oxides [16, 17], usually

described by dimmer models or related quantum ﬁeld theory models [18], seem to have anomalous critical

dynamics [19].

The standard approach to critical dynamics is the “Dynamical Renormalization Group (DRG)” [20],

distinctly developed in a seminal paper by Hohenberg and Halperin [21]. The simplest starting point is to

assume that, very near a critical point, the dynamics of the order parameter is governed by a dissipative

process driven by an overdamped additive noise Langevin equation. The typical relaxation time near a ﬁxed

point is given by τ∼ξz, where ξis the correlation length and zis the dynamical critical exponent. At a

critical point, ξ→ ∞ and therefore τ→ ∞, meaning that the system does not reach the equilibrium at

criticality. Together with usual static exponents, zdeﬁnes the universality class of the transition. Interest-

ingly, since the symmetry of the model does not constrain dynamics, there are diﬀerent dynamic universality

classes for the same critical point.

As usual in Renormalization Group (RG) theory [22], a RG transformation generates all kind of inter-

actions, compatible with the symmetry of the system. For this reason, a consistent study of a RG ﬂux

should begin, at least formally, with the most general Hamiltonian containing all couplings compatible with

symmetry. Interestingly, DRG transformations not only generate new couplings in the Hamiltonian, but also

modify the probability distribution of the stochastic process initially assumed. Thus, the stochastic noise

probability distribution also ﬂows with the DRG transformation, i.e., it is scale dependent. In particular,

we will show that one-loop perturbative corrections generate couplings compatible with multiplicative noise

stochastic processes, even in the case of assuming an additive dynamics as a starting condition. In order

to understand the fate of these couplings, we decided to analyze a more general dynamics for the order pa-

rameter near criticality. We assume a dissipative process driven by a multiplicative noise Langevin equation.

For concreteness, we address a simple model of a non conserved real scalar order parameter, ϕ(x, t) with

quartic coupling ϕ4(x, t) (model A of Ref. [21]). The dynamics is driven by a multiplicative noise Langevin

equation, characterized by a general dissipation function G(ϕ), with the same symmetry of the Hamiltonian.

Initially, we analyze the inﬂuence of a stochastic multiplicative evolution on the equilibrium properties

of the system. For this, we compute the asymptotic stationary probability distribution. We show that

the multiplicative diﬀusion function modiﬁes the asymptotic equilibrium potential. It is worth to note

that, depending on the parameters of the model, multiplicative dynamics could have dramatic eﬀects on

thermodynamics, possibly changing the order of the phase transition.

To study the dynamics of this problem, we implement a DRG using a functional formalism. For spatial

dimensions greater than the upper critical one (dc= 4 in the example discussed), the Gaussian ﬁxed point

controls the physics of the phase transition and all multiplicative noise couplings are irrelevant. Thus,

the assumption of an overdamped Langevin additive dynamics is correct. In this case, the static critical

exponents are provided as usual by a Landau theory and the dynamical critical exponent z= 2. However,

for d < 4, ﬂuctuations are important, the Gaussian ﬁxed point is unstable and the problem of the fate of

multiplicative couplings is more involved. We perform a systematic well controlled ϵ= 4 −dexpansion

of the DRG up to order ϵ2. We ﬁnd that, at order ϵ, the Wilson-Fisher ﬁxed point [23] shows up, the

multiplicative couplings are irrelevant and the dynamical critical exponent z= 2 + O(ϵ2), in agreement with

the results of Ref. [21]. However, at order ϵ2, the additive dynamics is no longer a ﬁxed point of the DRG

equations, giving place to a novel ﬁxed point with a true multiplicative dynamics, codiﬁed by the diﬀusion

function G(ϕ) = 1 + g∗ϕ2/2, where g∗∼ϵ2is a ﬁxed value of the coupling constant that characterizes the

multiplicative dynamics.

The paper is organized as follows: In §2 we make a brief review of multiplicative stochastic processes,

beginning with a single variable, passing through multiple variable systems, ending with extended continuous

systems appropriated to describe order parameters and phase transitions. In this section, we analyze in

detail equilibrium properties. In §3 we describe the functional formalism to describe dynamics of an order

parameter, presenting an speciﬁc model. In section 4, we built up the DRG equations and compute the

ﬂux diagram. Finally, we discuss our results in section 5. We leave calculation details for the appendices

Appendix A, Appendix B, Appendix C and Appendix D.

2. Brief review of multiplicative stochastic processes

To make the paper self-contained and to establish notation, we present in this section a very brief review

of multiplicative stochastic processes. Although the content of this section is not completely novel, we

believe that the way to present the subject clariﬁes some usual misunderstandings in the literature.

2.1. Single stochastic variable

Let us consider a single stochastic variable ϕ(t), whose dynamics is driven by the multiplicative Langevin

equation

dϕ(t)

dt =F[ϕ] + G(ϕ)η(t),(1)

where η(t) is a Gaussian white noise

⟨η(t)⟩= 0 ,(2)

⟨η(t)η(t′)⟩=β−1δ(t−t′).(3)

F(ϕ) is an arbitrary drift force and G(ϕ) is a diﬀusion function that characterizes the multiplicative stochastic

dynamics. The constant βmeasures the noise intensity and, under speciﬁc equilibrium conditions, can be

interpreted as a temperature parameter β= 1/kBT(kBis the Boltzmann constant).

In order to completely deﬁne Eq. (1), it is necessary to ﬁx the stochastic prescription necessary to deﬁne

the Wiener integrals. Along this paper we use the Generalized Stratonovich prescription, also known as

α-prescription, which is labeled by a real parameter 0 ≤α≤1. For concreteness, a solution of Eq. (1) will

contain integrals of the type ZG(ϕ(t))η(t)dt =ZG(ϕ(t))dW (4)

where W(t) is a Wiener process. By deﬁnition, the Riemann-Stieltjes integral is

ZG(ϕ(t))dW = lim

n→∞

i=1

G(ϕ(τi)) [W(ti+1)−W(ti)] (5)

where τiis taken in the interval [ti, ti+1]. The limit n→ ∞ is taken in the mean quadratic sense [1, 2].

For smooth measures W(t), the integral does not depend on τi. However, since it is a Wiener process, the

limit depends on the prescription to choose τi. In the Generalized Stratonovich prescription,τiis ﬁxed in

the following way:

G(ϕ(τi)) = G[(1 −α)ϕ(ti) + αϕ(ti+1)] (6)

where 0 ≤α≤1. In this way, α= 0 corresponds with the pre-point Itˆo prescription [24], α= 1/2

is the Stratonovich one [25], while α= 1 is known as the H¨anggi-Klimontovich or anti-Itˆo (post-point)

convention [26, 27]. Therefore, the dynamics described by equation 1 is completely determined by the drift

force F(ϕ), the diﬀusion function G(ϕ) and the stochastic prescription α.

It is possible to have a deeper insight by looking at the Fokker-Planck equation for the probability

distribution P(ϕ, t). It can be cast in the form of a continuity equation [28]

∂P (ϕ, t)

∂t +∂J(ϕ, t)

∂ϕ = 0 (7)

where the probability current is given by

J(ϕ, t) = F(ϕ)−(1 −α)β−1G(ϕ)G′(ϕ)P(ϕ, t)−1

2β−1G2(ϕ)∂P (ϕ, t)

∂ϕ (8)

in which G′=dG/dϕ. It is worth noting that for each value of α, we have diﬀerent dynamics. This eﬀect

is due to multiplicative noise. In fact, note that the term proportional to αis always multiplied by dG/dϕ.

Therefore, for additive noise, dG/dϕ = 0 and, as a consequence, the dynamics does not depend on the

stochastic prescription.

From equation (7), it is immediate to access the equilibrium properties of the stochastic system. Assuming

that, at long times, the system converges to a stationary state, the stationary probability

Pst(ϕ) = lim

t→∞ P(ϕ, t) (9)

satisﬁes, dJst

dϕ = 0 ,(10)

where Jst(ϕ) = limt→∞ J(ϕ, t) is given by equation (8), evaluated at the stationary probability Pst(ϕ). The

equilibrium solution of the Fokker-Planck equation satisﬁes Jst(ϕ) = 0. We immediately obtain

Peq(ϕ) = Ne−βUeq (ϕ)(11)

where Nis a normalization constant and

Ueq(ϕ) = −2ZϕF(x)

G2(x)dx + (1 −α)β−1ln G2(ϕ)(12)

Eqs. (11) and (12) codify the asymptotic equilibrium properties of the stochastic system driven by the

Langevin equation (1). The equilibrium properties are determined by both functions {F, G}and the stochas-

tic prescription α. If the stochastic system represents a conservative system in a noisy environment, i.e., if

it is possible to identify a deterministic system deﬁned by a potential or a Hamiltonian H(ϕ), then F(ϕ) is

time independent and can be written in terms of the Hamiltonian as

F(ϕ) = −1

2G2(ϕ)dH

dϕ .(13)

This equation is a generalization of the Einstein relation [28]. In this case, the equilibrium potential, Eq.

(12) takes the simpler form,

Ueq(ϕ) = H(ϕ) + (1 −α)

βln G2(ϕ)(14)

As expected, the equilibrium potential not only depends on the Hamiltonian and the diﬀusion function, but

also on the stochastic prescription α. Notice that, for α= 1, Ueq =Hand the probability distribution

is the Boltzmann distribution, as it should be for a thermodynamic physical system. For this reason, the

α= 1 prescription is also known as thermal prescription. For any other value of α, even the more usual

Stratonovich one, α= 1/2, the equilibrium potential is modiﬁed by the diﬀusion function G. This correction

is proportional to β−1, since it has a noisy origin.

An important consequence of Eq. (14) is that in a zero dimensional stochastic system (just one variable

ϕ(t)), the equilibrium state is always reached for almost any reasonable G(ϕ). The only condition is that

the equilibrium distribution should be normalizable, i.e.,N−1=Rdϕ exp{−βUeq}should be ﬁnite.

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TheroleofmultiplicativenoiseincriticaldynamicsNathanO.Silvanoa,b,∗,DanielG.BarcibaCenterforAdvancedSystemsUnderstanding,Untermarkt20,02826G¨orlitz,Helmholtz-ZentrumDresden-Rossendorf,BautznerLandstraße400,01328DresdenbDepartamentodeF´ısicaTe´orica,UniversidadedoEstadodoRiodeJaneiro,RuaS˜aoFranciscoX...

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