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Effects of temperature on the relaxation to equilibrium and stationary nonequilibrium states of some Langevin systems
ARTICLE IN PRESS
Physica A 374 (2007) 109–118 www.elsevier.com/locate/physa
Effects of temperature on the relaxation to equilibrium and stationary nonequilibrium states of some Langevin systems Humberto C.F. Lemos, Emmanuel Pereira Departamento de Fı´sica-ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil Received 17 November 2005; received in revised form 22 December 2005 Available online 31 July 2006
Abstract We investigate the effects of temperature on the properties of the time relaxation to equilibrium and nonequilibrium steady states of correlation functions of some Langevin harmonic systems. We consider commonly used dissipative and conservative Langevin dynamics, and show that the time relaxation rate depends on the temperature in the case of thermal reservoirs at different temperatures connected to the system, but it does not happen in the case of relaxation to equilibrium, i.e., if all the heat bath are at the same temperature. Our formalism maps the initial stochastic problem on a noncanonical quantum field theory, and the calculations of the relaxation rates are based on a perturbative analysis. We argue to show the reliability of the perturbative computation. r 2006 Elsevier B.V. All rights reserved. Keywords: Langevin systems; Time relaxation rate; Stationary states
1. Introduction The observable world is surrounded by phenomena involving nonequilibrium processes, but our understanding of such systems, i.e., the number of models that permit detailed calculations, is very limited [1]. Consequently, several attempts to extend central objects and relations of equilibrium statistical mechanics to nonequilibrium systems have been made, and works have been devoted to the study of stochastic dynamical systems describing nonequilibrium models from a microscopic point of view. To give some results, we recall e.g., the celebrated Gallavotti–Cohen theorem [2], which characterizes, for thermostated systems, the fluctuations of the entropy production in nonequilibrium statistical mechanics. In Ref. [3], we have another example of the stationary nonequilibrium state (SNS) characterization: it is constructed a free energy functional for the steady state of the open symmetric simple exclusion process, exhibiting a feature of macroscopically long range correlation, totally absent from equilibrium systems. A challenging problem related to the SNS characterization is the understanding of heat conduction in a lattice system with conservative dynamics, even in the 1D context [4,5]. The central issue is finding the conditions in a model Hamiltonian system which lead to the Fourier’s law. Almost all the results are obtained by means of computer Corresponding author.
E-mail addresses: hcfl@fisica.ufmg.br (H.C.F. Lemos), emmanuel@fisica.ufmg.br (E. Pereira). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.07.004
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simulations, and there are many contradictions in the literature [6–8]. In short, there are interesting and recent results, but there is a lack of unifying principles (compared to the equilibrium statistical physics), and it is unknown a simple way for seeking the properties of the steady states and the nonequilibrium phenomenology associated to the initial stochastic equations. It makes the detailed analysis of used dynamical models describing nonequilibrium processes a problem of general interest. In the present paper, we aim to study some nonequilibrium systems, starting also from a microscopic point of view (say, from some commonly used stochastic differential equations). We investigate the role played by temperature in the relaxation of the correlation functions to the steady states. We consider dissipative and conservative Langevin systems with exponential time relaxation and focus on the analysis of the properties of the relaxation rates for harmonic models (the investigation of harmonic systems is recurrent, see e.g., Refs. [9,10]), and we compare the cases of convergence to equilibrium and convergence to SNS. We first consider the stochastic lattice Ginzburg–Landau (GL) model with the (dissipative) dynamics given by a Langevin equation involving white noise with intensity changing in space, and we study the relaxation (time decay to the steady state) of fluctuations in the two-point function. Such a system may be interpreted, e.g., as d-dimensional crystal in contact with several thermal reservoirs, one at each site, at different temperatures (the reservoirs at the same temperature lead us to the particular case of convergence to equilibrium). For these systems, the equations of motion are dissipative: we have a model A in the terminology of Hohenberg and Halpering [11]. In sequel, we turn to the conservative Langevin dynamical systems, an important class of models, recurrent in the study of transport properties, heat flow, etc. In our approach, we construct an integral representation a la Feynman–Kac formalism, which relates the stochastic problem to a noncanonical quantum field theory. The formalism is suitable for the study of general correlation functions, in particular, for the study of their behavior (space and time decay) in the steady distribution. Let us list some results. We show, first for the dissipative systems, that the time relaxation rate does not depend on the value of the temperature if all the reservoirs are at the same temperature (i.e., for the case of convergence to equilibrium). For a temperature gradient among the reservoirs (and so, convergence to SNS), we show that, if a site is linked by the interaction to other sites connected to reservoirs where most of them are at higher temperature, then its relaxation rate is decreased (compared to the case of same temperature). Otherwise, the decay rate is increased. Similar results follow for the conservative systems. In short, we show that the relaxation rate to SNS of the correlation functions (even for harmonic models) depends on the temperature of the reservoirs connected to the system, but it does not happen in the case of relaxation to equilibrium. It is worth to emphasize that the correlation properties analyzed here are directly related to experimentally observable effects in concrete physical systems. For example, in the case of a magnetic system governed by the time-dependent GL model (the field variable describing the magnetization), our results show that the time relaxation of the magnetization fluctuations is sensitive to the temperature if the system is submitted to a temperature gradient, but it does not happen if the whole system is connected to a single thermal reservoir (i.e., to a unique temperature). Recently, several works [12–17] (and references there in) have been devoted to the detailed study of the relaxation properties of the two and/or four-point functions, for the easier case of dissipative dynamics and an unique thermal reservoir (i.e., convergence to equilibrium). The rest of the paper is organized as follows. In Section 2, we present the nonconservative model and analyze the case of a single temperature for the reservoirs. The integral formalism for the correlations of a system connected to reservoirs at different temperatures is developed in Section 3. In Section 4, we describe the perturbative analysis. Section 5 is devoted for the conservative models, Section 6 for their two-point function analysis, and Section 7 for the final remarks. 2. The nonconservative model and initial considerations Let us introduce the first model to be analyzed here. We consider the stochastic Langevin dynamics of some scalar field lattice models: precisely, we take a crystal, with unbounded continuous spin variables jð~ x; tÞ 2 R, ~ x in a lattice space box ~ x 2 L Zd in contact with Langevin-type heat bath at each site, in other words, with
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dynamics given by q 1 jð~ x; tÞ ¼ rSðjð~ x; tÞÞ þ Zð~ x; tÞ, qt 2 where rS ¼ dS=dj and the interaction S is ( ) X1 X 2 2 SðjÞ ¼ ½jð~ xÞJð~ x; ~ yÞjð~ yÞ þ m jð~ xÞ , 2 ~y2L ~ x2L
(1)
(2)
Jð~ x; ~ yÞ is some pair interaction with Jð~ x; ~ xÞ ¼ 0; Z is a family of Gaussian white-noise processes with the expectations EðZð~ x; tÞÞ ¼ 0;
EðZð~ x; tÞZð~ y; t0 ÞÞ ¼ g~x d~x;~y dðt t0 Þ,
(3)
x. The inclusion in S of an anharmonic on-site g~x 40 is the Pnoise strength, i.e., the heat bath temperature at site ~ potential ~x lPðjð~ xÞÞ gives us a general time-dependent Ginzburg–Landau model, a very well-known model which frequently appears in the study of dynamical critical phenomena [11]. To study the dynamics of these systems, we turn to the time evolution of functions f of the spin configuration ðjÞ defined by f t ðcÞ ¼ Eðf ðjðtÞÞÞ, where c ¼ jðt ¼ t0 Þ is some initial condition. Thus, starting from the Langevin equation (1) written as dj ¼ ½rS=2 dt þ g1=2 dB (where B is a Wiener process and g g~x is the intensity of the noise: note that g depends on ~ x only, not on j, and so, the noise is additive), the Itoˆ formula [18] gives us df ¼ rf g1=2 dB þ ð12 gr2 f 12 rS rf Þ dt, which leads to a Markov dynamics f t ðcÞ ¼ etH f ðcÞ, ( ) X 1 q2 1 qS q g~x þ f. Hf ¼ 2 xÞ qjð~ xÞ qjð~ xÞ2 2 qjð~ ~ x
ð4Þ
To proceed, we first analyze in detail the case of all reservoirs at the same temperature: g~x ¼ g. In this case, to give a concrete physical example, the nonconservative dynamics described now can be a good description of a magnetic system evolution to equilibrium, if the thermal conductivity of each reservoir is very high compared to that of the spins [11]. For such a situation, it follows that the generator of the dynamics H is a positive and Hermitian operator on L2 ðdmÞ, where dm eSðjÞ=g dj=normalization, i.e., Z f¯ ðjÞHf ðjÞ dmX0. ðf ; Hf ÞL2 ðdmÞ const: The positivity of H leads to an exponential convergence (time evolution) to 1, the ground state: the eigenfunction of H with zero eigenvalue. Thus, the distribution eSðjÞ=g dj=Z (Z is the normalization) in this space L2 ðdmÞ is the steady distribution. To study the time relaxation associated to H, we first note that, using the unitary operator U from L2 ðdmÞ to L2 ðdjÞ given by ðUf ÞðjÞ ¼ Z 1=2 eS=2g , we obtain the Schro¨dinger-type operator " # g X q2 1 X 1 qSL 2 q2 S L 1 H ¼ UHU ¼ þ 2 ~x2L qjð~ xÞ xÞ2 4 ~x2L 2g qjð~ qjð~ xÞ2 g X q2 1 X þ jð~ xÞ½ðJ þ m2 Þ2 jð~ xÞ þ c, ð5Þ ¼ 2 2 ~x2L qjð~ 8g ~x2L xÞ where c is an unimportant constant, related to d and m. Hence, we have a perfect connection with quantum theory, which gives us many tools and useful techniques to investigate the dynamics generator H. It is
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straightforward [19] to construct a representation a la Feynman–Kac formula for this dynamical problem. If f 1 ; . . . ; f n are functions of the spin configuration in L, OðjÞ ¼ 1 is the ground state of H L and t1 pt2 p ptn , then the Feynman–Kac formula is ðO; f 1 eðt2 t1 ÞH f 2 . . . eðtn tn1 ÞH f n OÞL2 ðdmÞ ¼ ðUO; f 1 eðt2 t1 ÞH f 2 . . . eðtn tn1 ÞH f n UOÞL2 ðdjÞ ¼
Z f 1 ðjðt1 ÞÞ . . . f n ðjðtn ÞÞ dn,
where dn is a Gaussian measure with correlations Z 1 Z Z X eip0 ðtt0 Þ ei~pð~x~yÞ g jð~ x; tÞ dn ¼ 0; jð~ x; tÞjð~ y; t0 Þ dn ¼ dp0 , 2 ~ pÞ þ m2 2 2pjLj 1 ~ p0 þ ½Jð~ 2 ~ p2L
ð6Þ
(7)
~ is its Fourier dual lattice; ~ ~ is a d-component vector ~ where jLj is the number of P points in L; L p2L p¼ ðp1 ; . . . ; pd Þ and ~ p ð~ x~ yÞ ¼ di¼1 pi ðxi yi Þ. We assumed that Jð~ x; ~ yÞ ¼ Jð~ x~ yÞ (i.e., translational invariance for J; in this paper we will also consider Jð~ x; ~ yÞ ¼ Jð~ y; ~ xÞ). We remark that, as we have a time convergence to equilibrium, i.e., for any initial state w wðtÞ ¼ etH wð0Þ!t!1 ðO; wÞO, the vacuum correlations above (6) shall give us the dominant time behavior of the correlations. x; tÞ, where To obtain the time relaxation rate for the two-point function S 2 ð~ Z x; tÞ ¼ S 2 ð~ x1 ~ x2 ; t1 t2 Þ ¼ S 2 ð~ x1 ; ~ x2 ; t1 ; t2 Þ ¼ jð~ x1 ; t1 Þjð~ x2 ; t2 Þ dn, S2 ð~ p; p0 Þ we turn to the singularities on the complex analytic continuation p0 ! ik0 of the Fourier transform S~ 2 ð~ ~ p; p0 Þ ¼ 1=S~ 2 ð~ (vide Paley–Wiener theorem [20]), or, which is simpler, for the zeros of Gð~ p; p0 Þ. The formula (7) gives us the following expressions for the zeros: 2 2 2 ~ pÞ þ m ~ pÞ þ m . 0 ¼ ðik0 Þ2 þ Jð~ (8) ) k0 ¼ Jð~ 2 2 ~ pÞXJð ~~ If J is a ferromagnetic-type interaction, i.e., Jð~ 0Þ, the relaxation rate will be given by ~~ M ¼ k0P ¼ Jð 0Þ þ m2 =2. E.g., for Jð~ x; ~ yÞ ¼ Dð~ x; ~ yÞ (D is the lattice Laplacian), we have ~ pÞ ¼ d ð1 cos pi Þ ) k0 ¼ m2 =2. Anyway, we have shown the feature that, for the considered Langevin Jð~ i¼1 dynamics, in the case of harmonic interaction and all the reservoirs at the same temperature, the time relaxation rate does not depend on the temperature. Our harmonic systems generalize the simplest example of such models, the well-known Ornstein–Uhlenbeck process: dvðtÞ ¼ xvðtÞ dt þ s dB, where the time relaxation rate depends only on the friction term, not on the temperature. However, we will see that it does not follow for the situation of the reservoirs at different temperatures. 3. Reservoirs at different temperatures The construction of a Feynman–Kac integral formalism for the correlation functions, as previously described in the case of time convergence to equilibrium, has been carried out using the positivity of the dynamics generator H in a properly chosen Hilbert space L2 ðdmÞ, namely, in a space with the appropriate Boltzmann weight. For the case of convergence to SNS, there is no simple prescription for the steady distribution, and we are forced to assume a new strategy. To analyze the stochastic dynamical system given by Eqs. (1)–(3), i.e., the model in contact with different heat baths, we start with the simpler case of J 0, which means that we turn off the coupling between sites, but we keep the different reservoirs. For J ¼ 0, any (decoupled) single site describes a system of one spin that goes to equilibrium (any site in a different temperature), and so, it is easy to obtain a formula for the correlations. In sequel, we introduce the coupling J and give the necessary corrections in the correlation integral formula using general ‘‘perturbation’’ theory of stochastic analysis, namely, using the Girsanov theorem (details below).
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For J 0, we have (using the notation j~x ðtÞ for jð~ x; tÞ) 1=2
dj~x ðtÞ ¼ 12 m2 j~x ðtÞ dt þ g~x dB~x ðtÞ,
(9)
whose solution is j~x ðtÞ ¼ e
ðm2 =2Þt
Z j~x ð0Þ þ 0
t
2
1=2
eðm =2ÞðtsÞ g~x dB~x ðsÞ,
(10)
which is a Gaussian process with expectations, taking for simplicity j~x ð0Þ ¼ 0, hj~x1 ðt1 Þj~x2 ðt2 Þi C 0 ð~ x1 ; t1 ; ~ x2 ; t2 Þ ¼
hj~x ðtÞi ¼ 0;
g~x1 d~x1 ;~x2 ðm2 =2Þjt1 t2 j e . m2
(11)
Introducing J we have 1=2
dj~x ðtÞ ¼ 12 m2 j~x ðtÞ dt þ h~x ðjÞ dt þ g~x dB~x ;
h~x ðjÞ ¼
1X J ~x;~y j~y . 2 ~y
(12)
The Girsanov theorem gives us a measure n for the new process as a ‘‘perturbation’’ of the Gaussian measure n0 , which comes from the first process where h 0. It states that nðAÞ ¼ E 0 ð1A ZðtÞÞ, where A is any measurable set, E 0 is the expectation for n0 ; 1A denotes the characteristic function; and considering the process starting at t0 ¼ 0, Z t Z 1 t 2 ~ uðjðsÞÞ djðsÞ kuðjðsÞÞk ds , ZðtÞ ¼ exp 2 0 0 ~ are given by where u and j g1=2 uðjÞ ¼ hðjÞ;
1=2
~ ~x ðtÞ ¼ dj~x ðtÞ þ 12 m2 j~x ðtÞ dt, g~x dj
the inner product and norm above are the canonical ones in RjLj . For a general statement of the Girsanov theorem see e.g., Ref. [18]; more comments and technical details for applications in related problems are given in Refs. [21,22]. We have # ) Z Z X( " 1X 1 2 1 g~x J ~x;~y j~y dj~x ðsÞ þ m j~x ðsÞ ds u dj~ ¼ 2 ~y 4 ~ x Z X X 1 1 2 m j~x ðsÞg~1 ¼ j~y J ~y;~x g~1 x;~ y j~ x ðsÞ y ds, x dj~ x J~ 2 2 ~ ~ x;~ y x;~ y and Z
kuk2 ds ¼
Z
1X j ðsÞJ ~x;~z g~1 z;~ y j~ y ðsÞ ds. z J~ 4 ~x;~y;~z ~x
Thus, turning to the integral formula for the correlations, we obtain R j~x ðt1 Þj~y ðt2 ÞZðtÞ dmC R hj~x ðt1 Þj~y ðt2 Þi ¼ , ZðtÞ dmC where t4t1 ; t2 ; C is the covariance (11) of the previous Gaussian process with h ¼ 0, and Z t W ðjðsÞÞ ds , ZðtÞ ¼ exp 0 2 X1 X1 m d j~x ðsÞJ ~x;~y g~y1 j~x ðsÞJ ~x;~z g~1 W ðjðsÞÞ ¼ þ j~y ðsÞ þ z;~ y j~ y ðsÞ. z J~ 2 dt 8 2 ~ ~ x;~ y x;~ y;~ z
(13)
ð14Þ
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4. Perturbative analysis We make a perturbative analysis in order to describe the dominant behavior of the two-point function. Supported by previous works (involving only convergence to equilibrium of anharmonic systems, but with an integral formalism with similar effective field theories), we claim that the perturbative analysis is correct: e.g., in Ref. [12], we prove, for the case of all reservoirs at the same low temperature g ¼ 15m2 (convergence to equilibrium) and with weak nonlinearity that the rigorous and complete treatment introduces only small corrections in the perturbative calculations (described in Ref. [13] and considering also the four-point correlation function). And for a similar nonconservative anharmonic system at high temperature, we developed a convergent P cluster expansion [14] which supports the perturbative analysis [15]. Let us take J ~x~y Jð~ x~ yÞ small, i.e., a weak coupling between the sites. Thus, for the two-point function R j~x ðt1 Þj~x ðt2 ÞZðtÞ dn0 ðjÞ R S2 ð~ x ; t1 ; ~ x; t 2 Þ ¼ , ZðtÞ dn0 ðjÞ after some (tedious, but straightforward) calculations, up to second order in J, we obtain ( ! ) m4 2 g~y g~x 1X 1 1 2 x 2 g~ 4 p0 ~ x ; p0 Þ ¼ m 4 1 ðJ ~x~y Þ þ ðJ Þ~x~x m4 , S2 ð~ 2 4 ~y g~y g~x m44 þ p20 2 ð 4 þ p20 Þ2 4 þ p0 x; p0 Þ is the (time only) Fourier transform of S 2 ð~ x; t1 t2 Þ S 2 ð~ x; t1 ; ~ x; t2 Þ. We may rewrite S~ 2 as where S~ 2 ð~ (again, considering up to second order in J) " ! #1 g~y m4 1 X x 2 g~ 2 ~ þ x; p0 Þ ¼ g~x p0 þ ðJ ~x~y Þ , (15) þ c2 S2 ð~ 4 ~y 4 g~y g~x where c2 involves the other factors of J above which do not depend on g. To get the time decay rate, we search for the singularities in S~ 2 for imaginary p0 ¼ ik0 . We obtain ! g~y m4 1X x 2 g~ 2 þ sðg~x Þ þ c2 ; sðg~x Þ ðJ ~x~y Þ M ~x ¼ , (16) 4 ~y 4 g~y g~x x; tÞ. The shift sðg~x Þ in the square of the ‘‘bare mass’’ m4 =4 above is the where M ~x is the time decay rate for S2 ð~ hallmark of the relaxation of a harmonic system with sites weakly coupled among themselves and each site connected to a different thermal reservoir. Note that s ¼ 0 if g~x g, i.e., all the reservoirs are at the same temperature. Let us analyze sðg~x Þ in some simple situations. For an unidimensional system (a bar) with Jðx; yÞ ¼ Dðx; yÞ if xay and Jðx; xÞ ¼ 0, i.e., coupling only between next-neighbor sites, and with the system submitted to a linear gradient of temperature: gx ¼ g0 þ cx, x ¼ 0; . . . ; N. The same follows, e.g., for a two-dimensional surface with g constant in one direction, and with a temperature gradient in the other one. We have (except for the sites at the ends of the bar): sðgx Þ ¼ 2 c2 =2ðg2x c2 Þ40. In short, the temperature gradient increases the relaxation rate. It does not happen for the first site of the bar, i.e., for site connected to the reservoir at lowest temperature: sðg0 Þo0. For the same system, e.g., a unidimensional bar, under a linear gradient of temperature gx ¼ g0 þ cx, but for Jðx; yÞ with long range, e.g., Jðx; yÞ ¼ if 0ojx yjpk, and 0 otherwise, where 1okoN, we have that sðgx Þ becomes negative in a region starting from the first site; the size of such a region depends on the gradient c and on the range of the interaction k. In short, the nonlocality in the interaction together with the temperature gradient extend the boundary effect and lead to a region where the relaxation rate (for the two-point fluctuation) is smaller than that for a system connected to reservoirs at the same temperature, and also to another (larger) region where the relaxation rate is increased. Again for a bar, with coupling between next-neighbor sites, connected to reservoirs with temperatures alternating between gþ and g , gþ 4g (e.g., the odd sites connected to gþ and the even ones connected to g ),
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we have sðgx Þ40 for the sites linked to gþ , and sðgx Þo0 for those linked to g (the end sites shall be analyzed in separate). Roughly, if a site (connected to its reservoir) is linked by the interaction to other sites connected to their reservoirs, most of them at higher temperature, then the site relaxation rate to SNS is decreased (the fluctuations persist for more time), compared to the relaxation for the case of all reservoirs at the same temperature. Otherwise, the decay rate is increased. 5. The conservative model Now we study the conservative dynamical model (as presented, e.g., in Ref. [9]) and derive the integral representation for the correlation functions, following the same steps from the previous sections. We consider a crystal with unbounded scalar variables in a d-dimensional lattice space box L Zd , with stochastic heat bath at each site. Precisely, we take a system of N harmonic oscillators with Hamiltonian Hðq; pÞ ¼
N N X 1 2 1 X ½pj þ Mq2j þ q J l;j qj , 2 2 jal¼1 l j¼1
(17)
where pj is the conjugate momentum with respect to qj , M40, and we take J l;l ¼ 0. The time evolution is given by the stochastic differential equations dqj ¼ pj dt; j ¼ 1; . . . ; N, qH 1=2 dt zpj dt þ gj dBj ; dpj ¼ qqj
j ¼ 1; . . . ; N,
ð18Þ
where Bj are independent Wiener processes, i.e., dBj =dt ¼ Zj ðtÞ are independent Gaussian white noises hZj ðtÞi ¼ 0;
hZj ðtÞZl ðt0 Þi ¼ gj dj;l dðt t0 Þ,
z40 is the heat bath coupling and gj ¼ 2zT j , where T j 40 is the temperature of the jth heat bath. To treat the dynamical system it is useful to introduce the phase-space vector f ¼ ðq; pÞ with 2N coordinates, and write the equation for the dynamics (18) as _ ¼ Af þ sZ, f
(19)
where Z is a 2N 1 column vector whose coordinates are independent white-noises (note that its contribution to f_ k is nonzero only for k4N), A and s are 2N 2N matrices, A ¼ A0 þ J and ! 0 0 0 I 0 0 pffiffiffiffiffiffiffiffiffiffiffi . A0 ¼ ; J¼ ; s¼ (20) 0 2GT M G J 0 The matrices above are composed by N N blocks: I is the unit N N matrix, and M ¼ MI, G ¼ zI and T is a diagonal matrix with Tjl ¼ T j djl . To study the dynamics we adopt the previous strategy used for the nonconservative models. First, we consider the system without coupling among the sites (i.e., with J 0). Then, each site of the lattice L is an independent system connected to a heat bath, and so, as t ! 1, we get equilibrium Gibbs distributions. To recover the original dynamical system, we introduce the coupling (interaction) among the sites and calculate the changes using the Girsanov theorem. The solution of (19) with J 0 is the Ornstein–Uhlenbeck process given by Ref. [17] Z t tA0 fðtÞ ¼ e fð0Þ þ ds eðtsÞA0 sZðsÞ. (21) 0
For simplicity we take fð0Þ ¼ 0. The covariance of this Gaussian process is ( eðtsÞA0 Cðs; sÞ; tXs; hfðtÞfðsÞi0 Cðt; sÞ ¼ T Cðt; tÞeðstÞA0 ; tps;
(22)
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Z
t
T
ds esA0 s2 esA0 .
Cðt; tÞ ¼
(23)
0
Diagonalizing A0 , it follows that 8 0 z > > I < I 0 B tanhðtrÞ B 2 expðtA0 Þ ¼ etðz=2Þ coshðtrÞ þ @ > r 0 I > M :
19 > I C> = C , A z > > I ; 2
(24)
where r ¼ ððz=2Þ2 MÞ1=2 . We have a similar expression for the transpose expðtAT0 Þ ¼ ½expðtA0 ÞT . If ðz=2Þ2 4M, r is real and 0oroz=2; otherwise, r is pure imaginary, but there is no problem (it does not spoil the dynamics): coshðtrÞ in the formula above becomes coshðt ir0 Þ ¼ cosðtr0 Þ, etc. As t ! 1, we have a convergence to equilibrium and the stationary state is Gaussian, with mean zero and covariance 0 1 Z 1 T T 0 A C¼ . (25) ds esA0 s2 esA0 ¼ @ M 0 0 T Now, we use the Girsanov theorem [18] to introduce (say, to turn on) the harmonic coupling potential. Then, as described in Section 3, we have a new measure given in terms of ZðtÞ, where Z t Z 1 t 2 1=2 ZðtÞ ¼ exp u dB u ds ; gi ui ¼ Ji;j fj ðtÞ. (26) 2 0 0 From (20) and the expression for ui , we have that ui is nonvanishing only for i4N (gj ¼ 0). From now on, we will use the following index notation: i 2 fN þ 1; N þ 2; . . . ; 2Ng, j 2 f1; 2; . . . ; Ng, and k 2 f1; 2; . . . ; 2Ng. Let us make explicit the terms in ZðtÞ. We have, for the first term, ui dBi ¼ g1 i Ji;j fj ðdfi þ ðA0 Þi;k fk dtÞ. For clearness, we rewrite the stochastic equations (18) for the decoupled process as dfj ¼ ðA0 Þjk fk dt ¼ fjþN dt, 1=2
dfi ¼ ðA0 Þik fk dt þ gi
dBi ¼ ðMfiN zfi Þ dt þ ð2zT iN Þ1=2 dBi .
ð27Þ
Hence, from Itoˆ formula [18] and Eq. (27), it follows that fj dfi ¼ dðfi fj Þ þ fi fjþN dt. For the u2 term we get 2 T 1 u2i ¼ g1 i ðJi;j fj Þ ¼ fj 0 Jj 0 ;i gi Ji;j fj .
(28)
Hence, for the correlation functions, we obtain (again) an integral representation involving a ‘‘perturbative’’ potential and a Gaussian measure: e.g., for the two-point function we have R fk1 ðt1 Þfk2 ðt2 ÞZðtÞ dmC ðfÞ R hfk1 ðt1 Þfk2 ðt2 Þi ¼ (29) ; t1 ; t2 ot; ZðtÞ dmC ðfÞ with Z t ZðtÞ exp½V ¼ expfF ðfðtÞÞ þ F ðfð0ÞÞg exp W ðfðsÞÞ ds , 0
(30)
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and F ðfðtÞÞ ¼ g1 i fi ðtÞJi;j fj ðtÞ 1 W ðfðsÞÞ ¼ g1 i fi ðsÞJi;j fjþN ðsÞ þ Mgi fiN ðsÞJi;j fj ðsÞ T 1 1 þ zg1 i fi ðsÞJi;j fj ðsÞ þ 2 fj 0 ðsÞJj 0 ;i gi Ji;j fj ðsÞ.
Recall that we are assuming the sum over i, j and j 0 in the expressions above.
6. The relaxation of the two-point function First we consider, very briefly, the case of relaxation to equilibrium, i.e., all the reservoirs at the same temperature. In this situation, the formulas for the covariance (22, 23), with A replacing A0 , show that the temperature, which comes from s2 , appears as a factor multiplying the whole expression: in other words, hfðtÞfðsÞi ¼ T Gðt; sÞ, where G does not depend on T. And so, the temperature does not have any effect on the relaxation rate. Now we turn to the analysis of the two-point correlation function for the system connected to different heat baths. In specific, we take hfu fv i with v 2 f1; . . . ; Ng and u ¼ v þ N: this correlation gives the heat flow (in the steady state) in a chain with harmonic interparticle potential (see Refs. [17,9]). We will also consider only the asymptotic behavior of large times, since the interesting feature is the behavior of the system close to the steady state: we take tbðt1 ; t2 Þb1. To simplify the computation, note that Cðt; sÞ, given by (22)–(25), may be written as ( eðtsÞA0 C þ OðeðtþsÞz=2 Þ if t4s; Cðt; sÞ ¼ T C eðstÞA0 þ OðeðtþsÞz=2 Þ if tos; and the second term above vanishes as t; t1 ; t2 ! 1. In order to evaluate the two-point function, we proceed as before: we consider a small interaction J coupling the sites and expand ZðtÞ up to second order in J (the first order expansion shows no dependence of the relaxation rate on different temperatures). The covariance of the decoupled conservative system (22)–(25) is more intricate than the covariance of the nonconservative model (11), but the computation is still possible. Performing the integrations and the time Fourier transform on t0 ¼ t1 t2 , we get T uN T v S~ 2 ðu; v; p0 Þ ¼ C 0 T v þ C 1 Ju;j JvþN;j T j þ C 2 Ji;uN Ji;v , T iN
(31)
(recall that u N ¼ v) where C 0 depends on M, z, r, p0 and J, but has no temperature dependence; C 1 and C 2 depends on M, z, r and p0 . If we take J T ¼ J, we obtain Ji;uN ¼ Ju;iN and Ji;v ¼ JvþN;i . In (31), the sum over the indices i and j is considered. Hence, using the latter expressions for J and factoring T v , we get Tj Tv 2 ~ S2 ðu; v; p0 Þ ¼ T v C 0 þ ðJu;j Þ C 1 þ C2 , Tv Tj which reminds (15), specially if we have C 1 ¼ C 2 (a huge computation is necessary to determine the exact relation between C 1 and C 2 ). Anyway, the (gradient) temperature dependence is clear, and properties similar to those already described for the previous nonconservative dynamics follow here. It is worth to emphasize, once more time, the reliability of the perturbative analysis: besides the previous comments on the nonconservative systems, we recall that, in the study of heat conduction for a harmonic chain with weak interactions (related to the study of a two-point function in the steady state of these conservative models), a perturbative analysis [17] gives the same result of the complete and rigorous treatment [9].
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7. Concluding remarks In order to search for basic properties of nonequilibrium Langevin systems, we analyze some harmonic models with dissipative and conservative dynamics, and investigate the time relaxation rate of fluctuations in their two-point function. We show that the time relaxation rate depends on the temperature in the case of fluctuations in the steady state of a system submitted to a temperature gradient, a property which does not exist in the case of convergence to equilibrium (i.e., for a system submitted to an unique temperature). In short, we present here a property of nonequilibrium statistical models which is absent from equilibrium systems. And, it is worth to recall, such property is directly related to observable effects in concrete physical systems. Finally, we stress that the perturbative calculation carried out within the integral formalism is not naive, as pointed out by the comparisons between rigorous and perturbative results recalled for the cases of dissipative and conservative dynamics. Acknowledgment Work partially supported by CNPq (Brazil). References [1] D. Ruelle, Nature 414 (2001) 263–264. [2] G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74 (1995) 2694–2697; G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80 (1995) 931–970. [3] B. Derrida, J.L. Lebowitz, E.R. Speer, Phys. Rev. Lett. 87 (2001) 150601. [4] F. Bonetto, J.L. Lebowitz, L. Rey-Bellet, in: A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinsk (Eds.), Mathematical Physics 2000, Imperial College Press, London, 2000, pp. 128–150. [5] S. Lepri, R. Livi, A. Politi, Phys. Rep. 377 (2003) 1–80. [6] O.V. Gendelman, A.V. Savin, Phys. Rev. Lett. 84 (2000) 2381–2384. [7] O.V. Gendelman, A.V. Savin, Phys. Rev. Lett. 94 (2005) 219405. [8] L. Yang, B. Hu, Phys. Rev. Lett. 94 (2005) 219404. [9] F. Bonetto, J.L. Lebowitz, J. Lukkarinen, J. Stat. Phys. 116 (2004) 783–813. [10] A. Dhar, Phys. Rev. Lett. 86 (2001) 3554–3557. [11] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435–479. [12] P.A. Faria da Veiga, M. O’Carroll, E. Pereira, R. Schor, Commun. Math. Phys. 220 (2001) 377–402. [13] R. Schor, J.C.A. Barata, P.A. Faria da Veiga, E. Pereira, Phys. Rev. E 59 (1999) 2689–2694. [14] R.S. Thebaldi, E. Pereira, A. Procacci, J. Math. Phys. 46 (2005) 53302. [15] E. Pereira, Phys. Rev. E 65 (2002) 56605. [16] E. Pereira, Physica D 192 (2004) 23–32. [17] E. Pereira, R. Falcao, Phys. Rev. E 70 (2004) 46105. [18] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 1991. [19] J. Glimm, A. Jaffe, Quantum Physics, Springer, New York, 1987. [20] M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [21] J. Dimock, J. Stat. Phys. 58 (1990) 1181–1207. [22] E. Pereira, Lett. Math. Phys. 64 (2003) 129–135.
Physica A 374 (2007) 109–118 www.elsevier.com/locate/physa
Effects of temperature on the relaxation to equilibrium and stationary nonequilibrium states of some Langevin systems Humberto C.F. Lemos, Emmanuel Pereira Departamento de Fı´sica-ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil Received 17 November 2005; received in revised form 22 December 2005 Available online 31 July 2006
Abstract We investigate the effects of temperature on the properties of the time relaxation to equilibrium and nonequilibrium steady states of correlation functions of some Langevin harmonic systems. We consider commonly used dissipative and conservative Langevin dynamics, and show that the time relaxation rate depends on the temperature in the case of thermal reservoirs at different temperatures connected to the system, but it does not happen in the case of relaxation to equilibrium, i.e., if all the heat bath are at the same temperature. Our formalism maps the initial stochastic problem on a noncanonical quantum field theory, and the calculations of the relaxation rates are based on a perturbative analysis. We argue to show the reliability of the perturbative computation. r 2006 Elsevier B.V. All rights reserved. Keywords: Langevin systems; Time relaxation rate; Stationary states
1. Introduction The observable world is surrounded by phenomena involving nonequilibrium processes, but our understanding of such systems, i.e., the number of models that permit detailed calculations, is very limited [1]. Consequently, several attempts to extend central objects and relations of equilibrium statistical mechanics to nonequilibrium systems have been made, and works have been devoted to the study of stochastic dynamical systems describing nonequilibrium models from a microscopic point of view. To give some results, we recall e.g., the celebrated Gallavotti–Cohen theorem [2], which characterizes, for thermostated systems, the fluctuations of the entropy production in nonequilibrium statistical mechanics. In Ref. [3], we have another example of the stationary nonequilibrium state (SNS) characterization: it is constructed a free energy functional for the steady state of the open symmetric simple exclusion process, exhibiting a feature of macroscopically long range correlation, totally absent from equilibrium systems. A challenging problem related to the SNS characterization is the understanding of heat conduction in a lattice system with conservative dynamics, even in the 1D context [4,5]. The central issue is finding the conditions in a model Hamiltonian system which lead to the Fourier’s law. Almost all the results are obtained by means of computer Corresponding author.
E-mail addresses: hcfl@fisica.ufmg.br (H.C.F. Lemos), emmanuel@fisica.ufmg.br (E. Pereira). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.07.004
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simulations, and there are many contradictions in the literature [6–8]. In short, there are interesting and recent results, but there is a lack of unifying principles (compared to the equilibrium statistical physics), and it is unknown a simple way for seeking the properties of the steady states and the nonequilibrium phenomenology associated to the initial stochastic equations. It makes the detailed analysis of used dynamical models describing nonequilibrium processes a problem of general interest. In the present paper, we aim to study some nonequilibrium systems, starting also from a microscopic point of view (say, from some commonly used stochastic differential equations). We investigate the role played by temperature in the relaxation of the correlation functions to the steady states. We consider dissipative and conservative Langevin systems with exponential time relaxation and focus on the analysis of the properties of the relaxation rates for harmonic models (the investigation of harmonic systems is recurrent, see e.g., Refs. [9,10]), and we compare the cases of convergence to equilibrium and convergence to SNS. We first consider the stochastic lattice Ginzburg–Landau (GL) model with the (dissipative) dynamics given by a Langevin equation involving white noise with intensity changing in space, and we study the relaxation (time decay to the steady state) of fluctuations in the two-point function. Such a system may be interpreted, e.g., as d-dimensional crystal in contact with several thermal reservoirs, one at each site, at different temperatures (the reservoirs at the same temperature lead us to the particular case of convergence to equilibrium). For these systems, the equations of motion are dissipative: we have a model A in the terminology of Hohenberg and Halpering [11]. In sequel, we turn to the conservative Langevin dynamical systems, an important class of models, recurrent in the study of transport properties, heat flow, etc. In our approach, we construct an integral representation a la Feynman–Kac formalism, which relates the stochastic problem to a noncanonical quantum field theory. The formalism is suitable for the study of general correlation functions, in particular, for the study of their behavior (space and time decay) in the steady distribution. Let us list some results. We show, first for the dissipative systems, that the time relaxation rate does not depend on the value of the temperature if all the reservoirs are at the same temperature (i.e., for the case of convergence to equilibrium). For a temperature gradient among the reservoirs (and so, convergence to SNS), we show that, if a site is linked by the interaction to other sites connected to reservoirs where most of them are at higher temperature, then its relaxation rate is decreased (compared to the case of same temperature). Otherwise, the decay rate is increased. Similar results follow for the conservative systems. In short, we show that the relaxation rate to SNS of the correlation functions (even for harmonic models) depends on the temperature of the reservoirs connected to the system, but it does not happen in the case of relaxation to equilibrium. It is worth to emphasize that the correlation properties analyzed here are directly related to experimentally observable effects in concrete physical systems. For example, in the case of a magnetic system governed by the time-dependent GL model (the field variable describing the magnetization), our results show that the time relaxation of the magnetization fluctuations is sensitive to the temperature if the system is submitted to a temperature gradient, but it does not happen if the whole system is connected to a single thermal reservoir (i.e., to a unique temperature). Recently, several works [12–17] (and references there in) have been devoted to the detailed study of the relaxation properties of the two and/or four-point functions, for the easier case of dissipative dynamics and an unique thermal reservoir (i.e., convergence to equilibrium). The rest of the paper is organized as follows. In Section 2, we present the nonconservative model and analyze the case of a single temperature for the reservoirs. The integral formalism for the correlations of a system connected to reservoirs at different temperatures is developed in Section 3. In Section 4, we describe the perturbative analysis. Section 5 is devoted for the conservative models, Section 6 for their two-point function analysis, and Section 7 for the final remarks. 2. The nonconservative model and initial considerations Let us introduce the first model to be analyzed here. We consider the stochastic Langevin dynamics of some scalar field lattice models: precisely, we take a crystal, with unbounded continuous spin variables jð~ x; tÞ 2 R, ~ x in a lattice space box ~ x 2 L Zd in contact with Langevin-type heat bath at each site, in other words, with
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dynamics given by q 1 jð~ x; tÞ ¼ rSðjð~ x; tÞÞ þ Zð~ x; tÞ, qt 2 where rS ¼ dS=dj and the interaction S is ( ) X1 X 2 2 SðjÞ ¼ ½jð~ xÞJð~ x; ~ yÞjð~ yÞ þ m jð~ xÞ , 2 ~y2L ~ x2L
(1)
(2)
Jð~ x; ~ yÞ is some pair interaction with Jð~ x; ~ xÞ ¼ 0; Z is a family of Gaussian white-noise processes with the expectations EðZð~ x; tÞÞ ¼ 0;
EðZð~ x; tÞZð~ y; t0 ÞÞ ¼ g~x d~x;~y dðt t0 Þ,
(3)
x. The inclusion in S of an anharmonic on-site g~x 40 is the Pnoise strength, i.e., the heat bath temperature at site ~ potential ~x lPðjð~ xÞÞ gives us a general time-dependent Ginzburg–Landau model, a very well-known model which frequently appears in the study of dynamical critical phenomena [11]. To study the dynamics of these systems, we turn to the time evolution of functions f of the spin configuration ðjÞ defined by f t ðcÞ ¼ Eðf ðjðtÞÞÞ, where c ¼ jðt ¼ t0 Þ is some initial condition. Thus, starting from the Langevin equation (1) written as dj ¼ ½rS=2 dt þ g1=2 dB (where B is a Wiener process and g g~x is the intensity of the noise: note that g depends on ~ x only, not on j, and so, the noise is additive), the Itoˆ formula [18] gives us df ¼ rf g1=2 dB þ ð12 gr2 f 12 rS rf Þ dt, which leads to a Markov dynamics f t ðcÞ ¼ etH f ðcÞ, ( ) X 1 q2 1 qS q g~x þ f. Hf ¼ 2 xÞ qjð~ xÞ qjð~ xÞ2 2 qjð~ ~ x
ð4Þ
To proceed, we first analyze in detail the case of all reservoirs at the same temperature: g~x ¼ g. In this case, to give a concrete physical example, the nonconservative dynamics described now can be a good description of a magnetic system evolution to equilibrium, if the thermal conductivity of each reservoir is very high compared to that of the spins [11]. For such a situation, it follows that the generator of the dynamics H is a positive and Hermitian operator on L2 ðdmÞ, where dm eSðjÞ=g dj=normalization, i.e., Z f¯ ðjÞHf ðjÞ dmX0. ðf ; Hf ÞL2 ðdmÞ const: The positivity of H leads to an exponential convergence (time evolution) to 1, the ground state: the eigenfunction of H with zero eigenvalue. Thus, the distribution eSðjÞ=g dj=Z (Z is the normalization) in this space L2 ðdmÞ is the steady distribution. To study the time relaxation associated to H, we first note that, using the unitary operator U from L2 ðdmÞ to L2 ðdjÞ given by ðUf ÞðjÞ ¼ Z 1=2 eS=2g , we obtain the Schro¨dinger-type operator " # g X q2 1 X 1 qSL 2 q2 S L 1 H ¼ UHU ¼ þ 2 ~x2L qjð~ xÞ xÞ2 4 ~x2L 2g qjð~ qjð~ xÞ2 g X q2 1 X þ jð~ xÞ½ðJ þ m2 Þ2 jð~ xÞ þ c, ð5Þ ¼ 2 2 ~x2L qjð~ 8g ~x2L xÞ where c is an unimportant constant, related to d and m. Hence, we have a perfect connection with quantum theory, which gives us many tools and useful techniques to investigate the dynamics generator H. It is
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straightforward [19] to construct a representation a la Feynman–Kac formula for this dynamical problem. If f 1 ; . . . ; f n are functions of the spin configuration in L, OðjÞ ¼ 1 is the ground state of H L and t1 pt2 p ptn , then the Feynman–Kac formula is ðO; f 1 eðt2 t1 ÞH f 2 . . . eðtn tn1 ÞH f n OÞL2 ðdmÞ ¼ ðUO; f 1 eðt2 t1 ÞH f 2 . . . eðtn tn1 ÞH f n UOÞL2 ðdjÞ ¼
Z f 1 ðjðt1 ÞÞ . . . f n ðjðtn ÞÞ dn,
where dn is a Gaussian measure with correlations Z 1 Z Z X eip0 ðtt0 Þ ei~pð~x~yÞ g jð~ x; tÞ dn ¼ 0; jð~ x; tÞjð~ y; t0 Þ dn ¼ dp0 , 2 ~ pÞ þ m2 2 2pjLj 1 ~ p0 þ ½Jð~ 2 ~ p2L
ð6Þ
(7)
~ is its Fourier dual lattice; ~ ~ is a d-component vector ~ where jLj is the number of P points in L; L p2L p¼ ðp1 ; . . . ; pd Þ and ~ p ð~ x~ yÞ ¼ di¼1 pi ðxi yi Þ. We assumed that Jð~ x; ~ yÞ ¼ Jð~ x~ yÞ (i.e., translational invariance for J; in this paper we will also consider Jð~ x; ~ yÞ ¼ Jð~ y; ~ xÞ). We remark that, as we have a time convergence to equilibrium, i.e., for any initial state w wðtÞ ¼ etH wð0Þ!t!1 ðO; wÞO, the vacuum correlations above (6) shall give us the dominant time behavior of the correlations. x; tÞ, where To obtain the time relaxation rate for the two-point function S 2 ð~ Z x; tÞ ¼ S 2 ð~ x1 ~ x2 ; t1 t2 Þ ¼ S 2 ð~ x1 ; ~ x2 ; t1 ; t2 Þ ¼ jð~ x1 ; t1 Þjð~ x2 ; t2 Þ dn, S2 ð~ p; p0 Þ we turn to the singularities on the complex analytic continuation p0 ! ik0 of the Fourier transform S~ 2 ð~ ~ p; p0 Þ ¼ 1=S~ 2 ð~ (vide Paley–Wiener theorem [20]), or, which is simpler, for the zeros of Gð~ p; p0 Þ. The formula (7) gives us the following expressions for the zeros: 2 2 2 ~ pÞ þ m ~ pÞ þ m . 0 ¼ ðik0 Þ2 þ Jð~ (8) ) k0 ¼ Jð~ 2 2 ~ pÞXJð ~~ If J is a ferromagnetic-type interaction, i.e., Jð~ 0Þ, the relaxation rate will be given by ~~ M ¼ k0P ¼ Jð 0Þ þ m2 =2. E.g., for Jð~ x; ~ yÞ ¼ Dð~ x; ~ yÞ (D is the lattice Laplacian), we have ~ pÞ ¼ d ð1 cos pi Þ ) k0 ¼ m2 =2. Anyway, we have shown the feature that, for the considered Langevin Jð~ i¼1 dynamics, in the case of harmonic interaction and all the reservoirs at the same temperature, the time relaxation rate does not depend on the temperature. Our harmonic systems generalize the simplest example of such models, the well-known Ornstein–Uhlenbeck process: dvðtÞ ¼ xvðtÞ dt þ s dB, where the time relaxation rate depends only on the friction term, not on the temperature. However, we will see that it does not follow for the situation of the reservoirs at different temperatures. 3. Reservoirs at different temperatures The construction of a Feynman–Kac integral formalism for the correlation functions, as previously described in the case of time convergence to equilibrium, has been carried out using the positivity of the dynamics generator H in a properly chosen Hilbert space L2 ðdmÞ, namely, in a space with the appropriate Boltzmann weight. For the case of convergence to SNS, there is no simple prescription for the steady distribution, and we are forced to assume a new strategy. To analyze the stochastic dynamical system given by Eqs. (1)–(3), i.e., the model in contact with different heat baths, we start with the simpler case of J 0, which means that we turn off the coupling between sites, but we keep the different reservoirs. For J ¼ 0, any (decoupled) single site describes a system of one spin that goes to equilibrium (any site in a different temperature), and so, it is easy to obtain a formula for the correlations. In sequel, we introduce the coupling J and give the necessary corrections in the correlation integral formula using general ‘‘perturbation’’ theory of stochastic analysis, namely, using the Girsanov theorem (details below).
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For J 0, we have (using the notation j~x ðtÞ for jð~ x; tÞ) 1=2
dj~x ðtÞ ¼ 12 m2 j~x ðtÞ dt þ g~x dB~x ðtÞ,
(9)
whose solution is j~x ðtÞ ¼ e
ðm2 =2Þt
Z j~x ð0Þ þ 0
t
2
1=2
eðm =2ÞðtsÞ g~x dB~x ðsÞ,
(10)
which is a Gaussian process with expectations, taking for simplicity j~x ð0Þ ¼ 0, hj~x1 ðt1 Þj~x2 ðt2 Þi C 0 ð~ x1 ; t1 ; ~ x2 ; t2 Þ ¼
hj~x ðtÞi ¼ 0;
g~x1 d~x1 ;~x2 ðm2 =2Þjt1 t2 j e . m2
(11)
Introducing J we have 1=2
dj~x ðtÞ ¼ 12 m2 j~x ðtÞ dt þ h~x ðjÞ dt þ g~x dB~x ;
h~x ðjÞ ¼
1X J ~x;~y j~y . 2 ~y
(12)
The Girsanov theorem gives us a measure n for the new process as a ‘‘perturbation’’ of the Gaussian measure n0 , which comes from the first process where h 0. It states that nðAÞ ¼ E 0 ð1A ZðtÞÞ, where A is any measurable set, E 0 is the expectation for n0 ; 1A denotes the characteristic function; and considering the process starting at t0 ¼ 0, Z t Z 1 t 2 ~ uðjðsÞÞ djðsÞ kuðjðsÞÞk ds , ZðtÞ ¼ exp 2 0 0 ~ are given by where u and j g1=2 uðjÞ ¼ hðjÞ;
1=2
~ ~x ðtÞ ¼ dj~x ðtÞ þ 12 m2 j~x ðtÞ dt, g~x dj
the inner product and norm above are the canonical ones in RjLj . For a general statement of the Girsanov theorem see e.g., Ref. [18]; more comments and technical details for applications in related problems are given in Refs. [21,22]. We have # ) Z Z X( " 1X 1 2 1 g~x J ~x;~y j~y dj~x ðsÞ þ m j~x ðsÞ ds u dj~ ¼ 2 ~y 4 ~ x Z X X 1 1 2 m j~x ðsÞg~1 ¼ j~y J ~y;~x g~1 x;~ y j~ x ðsÞ y ds, x dj~ x J~ 2 2 ~ ~ x;~ y x;~ y and Z
kuk2 ds ¼
Z
1X j ðsÞJ ~x;~z g~1 z;~ y j~ y ðsÞ ds. z J~ 4 ~x;~y;~z ~x
Thus, turning to the integral formula for the correlations, we obtain R j~x ðt1 Þj~y ðt2 ÞZðtÞ dmC R hj~x ðt1 Þj~y ðt2 Þi ¼ , ZðtÞ dmC where t4t1 ; t2 ; C is the covariance (11) of the previous Gaussian process with h ¼ 0, and Z t W ðjðsÞÞ ds , ZðtÞ ¼ exp 0 2 X1 X1 m d j~x ðsÞJ ~x;~y g~y1 j~x ðsÞJ ~x;~z g~1 W ðjðsÞÞ ¼ þ j~y ðsÞ þ z;~ y j~ y ðsÞ. z J~ 2 dt 8 2 ~ ~ x;~ y x;~ y;~ z
(13)
ð14Þ
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4. Perturbative analysis We make a perturbative analysis in order to describe the dominant behavior of the two-point function. Supported by previous works (involving only convergence to equilibrium of anharmonic systems, but with an integral formalism with similar effective field theories), we claim that the perturbative analysis is correct: e.g., in Ref. [12], we prove, for the case of all reservoirs at the same low temperature g ¼ 15m2 (convergence to equilibrium) and with weak nonlinearity that the rigorous and complete treatment introduces only small corrections in the perturbative calculations (described in Ref. [13] and considering also the four-point correlation function). And for a similar nonconservative anharmonic system at high temperature, we developed a convergent P cluster expansion [14] which supports the perturbative analysis [15]. Let us take J ~x~y Jð~ x~ yÞ small, i.e., a weak coupling between the sites. Thus, for the two-point function R j~x ðt1 Þj~x ðt2 ÞZðtÞ dn0 ðjÞ R S2 ð~ x ; t1 ; ~ x; t 2 Þ ¼ , ZðtÞ dn0 ðjÞ after some (tedious, but straightforward) calculations, up to second order in J, we obtain ( ! ) m4 2 g~y g~x 1X 1 1 2 x 2 g~ 4 p0 ~ x ; p0 Þ ¼ m 4 1 ðJ ~x~y Þ þ ðJ Þ~x~x m4 , S2 ð~ 2 4 ~y g~y g~x m44 þ p20 2 ð 4 þ p20 Þ2 4 þ p0 x; p0 Þ is the (time only) Fourier transform of S 2 ð~ x; t1 t2 Þ S 2 ð~ x; t1 ; ~ x; t2 Þ. We may rewrite S~ 2 as where S~ 2 ð~ (again, considering up to second order in J) " ! #1 g~y m4 1 X x 2 g~ 2 ~ þ x; p0 Þ ¼ g~x p0 þ ðJ ~x~y Þ , (15) þ c2 S2 ð~ 4 ~y 4 g~y g~x where c2 involves the other factors of J above which do not depend on g. To get the time decay rate, we search for the singularities in S~ 2 for imaginary p0 ¼ ik0 . We obtain ! g~y m4 1X x 2 g~ 2 þ sðg~x Þ þ c2 ; sðg~x Þ ðJ ~x~y Þ M ~x ¼ , (16) 4 ~y 4 g~y g~x x; tÞ. The shift sðg~x Þ in the square of the ‘‘bare mass’’ m4 =4 above is the where M ~x is the time decay rate for S2 ð~ hallmark of the relaxation of a harmonic system with sites weakly coupled among themselves and each site connected to a different thermal reservoir. Note that s ¼ 0 if g~x g, i.e., all the reservoirs are at the same temperature. Let us analyze sðg~x Þ in some simple situations. For an unidimensional system (a bar) with Jðx; yÞ ¼ Dðx; yÞ if xay and Jðx; xÞ ¼ 0, i.e., coupling only between next-neighbor sites, and with the system submitted to a linear gradient of temperature: gx ¼ g0 þ cx, x ¼ 0; . . . ; N. The same follows, e.g., for a two-dimensional surface with g constant in one direction, and with a temperature gradient in the other one. We have (except for the sites at the ends of the bar): sðgx Þ ¼ 2 c2 =2ðg2x c2 Þ40. In short, the temperature gradient increases the relaxation rate. It does not happen for the first site of the bar, i.e., for site connected to the reservoir at lowest temperature: sðg0 Þo0. For the same system, e.g., a unidimensional bar, under a linear gradient of temperature gx ¼ g0 þ cx, but for Jðx; yÞ with long range, e.g., Jðx; yÞ ¼ if 0ojx yjpk, and 0 otherwise, where 1okoN, we have that sðgx Þ becomes negative in a region starting from the first site; the size of such a region depends on the gradient c and on the range of the interaction k. In short, the nonlocality in the interaction together with the temperature gradient extend the boundary effect and lead to a region where the relaxation rate (for the two-point fluctuation) is smaller than that for a system connected to reservoirs at the same temperature, and also to another (larger) region where the relaxation rate is increased. Again for a bar, with coupling between next-neighbor sites, connected to reservoirs with temperatures alternating between gþ and g , gþ 4g (e.g., the odd sites connected to gþ and the even ones connected to g ),
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we have sðgx Þ40 for the sites linked to gþ , and sðgx Þo0 for those linked to g (the end sites shall be analyzed in separate). Roughly, if a site (connected to its reservoir) is linked by the interaction to other sites connected to their reservoirs, most of them at higher temperature, then the site relaxation rate to SNS is decreased (the fluctuations persist for more time), compared to the relaxation for the case of all reservoirs at the same temperature. Otherwise, the decay rate is increased. 5. The conservative model Now we study the conservative dynamical model (as presented, e.g., in Ref. [9]) and derive the integral representation for the correlation functions, following the same steps from the previous sections. We consider a crystal with unbounded scalar variables in a d-dimensional lattice space box L Zd , with stochastic heat bath at each site. Precisely, we take a system of N harmonic oscillators with Hamiltonian Hðq; pÞ ¼
N N X 1 2 1 X ½pj þ Mq2j þ q J l;j qj , 2 2 jal¼1 l j¼1
(17)
where pj is the conjugate momentum with respect to qj , M40, and we take J l;l ¼ 0. The time evolution is given by the stochastic differential equations dqj ¼ pj dt; j ¼ 1; . . . ; N, qH 1=2 dt zpj dt þ gj dBj ; dpj ¼ qqj
j ¼ 1; . . . ; N,
ð18Þ
where Bj are independent Wiener processes, i.e., dBj =dt ¼ Zj ðtÞ are independent Gaussian white noises hZj ðtÞi ¼ 0;
hZj ðtÞZl ðt0 Þi ¼ gj dj;l dðt t0 Þ,
z40 is the heat bath coupling and gj ¼ 2zT j , where T j 40 is the temperature of the jth heat bath. To treat the dynamical system it is useful to introduce the phase-space vector f ¼ ðq; pÞ with 2N coordinates, and write the equation for the dynamics (18) as _ ¼ Af þ sZ, f
(19)
where Z is a 2N 1 column vector whose coordinates are independent white-noises (note that its contribution to f_ k is nonzero only for k4N), A and s are 2N 2N matrices, A ¼ A0 þ J and ! 0 0 0 I 0 0 pffiffiffiffiffiffiffiffiffiffiffi . A0 ¼ ; J¼ ; s¼ (20) 0 2GT M G J 0 The matrices above are composed by N N blocks: I is the unit N N matrix, and M ¼ MI, G ¼ zI and T is a diagonal matrix with Tjl ¼ T j djl . To study the dynamics we adopt the previous strategy used for the nonconservative models. First, we consider the system without coupling among the sites (i.e., with J 0). Then, each site of the lattice L is an independent system connected to a heat bath, and so, as t ! 1, we get equilibrium Gibbs distributions. To recover the original dynamical system, we introduce the coupling (interaction) among the sites and calculate the changes using the Girsanov theorem. The solution of (19) with J 0 is the Ornstein–Uhlenbeck process given by Ref. [17] Z t tA0 fðtÞ ¼ e fð0Þ þ ds eðtsÞA0 sZðsÞ. (21) 0
For simplicity we take fð0Þ ¼ 0. The covariance of this Gaussian process is ( eðtsÞA0 Cðs; sÞ; tXs; hfðtÞfðsÞi0 Cðt; sÞ ¼ T Cðt; tÞeðstÞA0 ; tps;
(22)
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Z
t
T
ds esA0 s2 esA0 .
Cðt; tÞ ¼
(23)
0
Diagonalizing A0 , it follows that 8 0 z > > I < I 0 B tanhðtrÞ B 2 expðtA0 Þ ¼ etðz=2Þ coshðtrÞ þ @ > r 0 I > M :
19 > I C> = C , A z > > I ; 2
(24)
where r ¼ ððz=2Þ2 MÞ1=2 . We have a similar expression for the transpose expðtAT0 Þ ¼ ½expðtA0 ÞT . If ðz=2Þ2 4M, r is real and 0oroz=2; otherwise, r is pure imaginary, but there is no problem (it does not spoil the dynamics): coshðtrÞ in the formula above becomes coshðt ir0 Þ ¼ cosðtr0 Þ, etc. As t ! 1, we have a convergence to equilibrium and the stationary state is Gaussian, with mean zero and covariance 0 1 Z 1 T T 0 A C¼ . (25) ds esA0 s2 esA0 ¼ @ M 0 0 T Now, we use the Girsanov theorem [18] to introduce (say, to turn on) the harmonic coupling potential. Then, as described in Section 3, we have a new measure given in terms of ZðtÞ, where Z t Z 1 t 2 1=2 ZðtÞ ¼ exp u dB u ds ; gi ui ¼ Ji;j fj ðtÞ. (26) 2 0 0 From (20) and the expression for ui , we have that ui is nonvanishing only for i4N (gj ¼ 0). From now on, we will use the following index notation: i 2 fN þ 1; N þ 2; . . . ; 2Ng, j 2 f1; 2; . . . ; Ng, and k 2 f1; 2; . . . ; 2Ng. Let us make explicit the terms in ZðtÞ. We have, for the first term, ui dBi ¼ g1 i Ji;j fj ðdfi þ ðA0 Þi;k fk dtÞ. For clearness, we rewrite the stochastic equations (18) for the decoupled process as dfj ¼ ðA0 Þjk fk dt ¼ fjþN dt, 1=2
dfi ¼ ðA0 Þik fk dt þ gi
dBi ¼ ðMfiN zfi Þ dt þ ð2zT iN Þ1=2 dBi .
ð27Þ
Hence, from Itoˆ formula [18] and Eq. (27), it follows that fj dfi ¼ dðfi fj Þ þ fi fjþN dt. For the u2 term we get 2 T 1 u2i ¼ g1 i ðJi;j fj Þ ¼ fj 0 Jj 0 ;i gi Ji;j fj .
(28)
Hence, for the correlation functions, we obtain (again) an integral representation involving a ‘‘perturbative’’ potential and a Gaussian measure: e.g., for the two-point function we have R fk1 ðt1 Þfk2 ðt2 ÞZðtÞ dmC ðfÞ R hfk1 ðt1 Þfk2 ðt2 Þi ¼ (29) ; t1 ; t2 ot; ZðtÞ dmC ðfÞ with Z t ZðtÞ exp½V ¼ expfF ðfðtÞÞ þ F ðfð0ÞÞg exp W ðfðsÞÞ ds , 0
(30)
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and F ðfðtÞÞ ¼ g1 i fi ðtÞJi;j fj ðtÞ 1 W ðfðsÞÞ ¼ g1 i fi ðsÞJi;j fjþN ðsÞ þ Mgi fiN ðsÞJi;j fj ðsÞ T 1 1 þ zg1 i fi ðsÞJi;j fj ðsÞ þ 2 fj 0 ðsÞJj 0 ;i gi Ji;j fj ðsÞ.
Recall that we are assuming the sum over i, j and j 0 in the expressions above.
6. The relaxation of the two-point function First we consider, very briefly, the case of relaxation to equilibrium, i.e., all the reservoirs at the same temperature. In this situation, the formulas for the covariance (22, 23), with A replacing A0 , show that the temperature, which comes from s2 , appears as a factor multiplying the whole expression: in other words, hfðtÞfðsÞi ¼ T Gðt; sÞ, where G does not depend on T. And so, the temperature does not have any effect on the relaxation rate. Now we turn to the analysis of the two-point correlation function for the system connected to different heat baths. In specific, we take hfu fv i with v 2 f1; . . . ; Ng and u ¼ v þ N: this correlation gives the heat flow (in the steady state) in a chain with harmonic interparticle potential (see Refs. [17,9]). We will also consider only the asymptotic behavior of large times, since the interesting feature is the behavior of the system close to the steady state: we take tbðt1 ; t2 Þb1. To simplify the computation, note that Cðt; sÞ, given by (22)–(25), may be written as ( eðtsÞA0 C þ OðeðtþsÞz=2 Þ if t4s; Cðt; sÞ ¼ T C eðstÞA0 þ OðeðtþsÞz=2 Þ if tos; and the second term above vanishes as t; t1 ; t2 ! 1. In order to evaluate the two-point function, we proceed as before: we consider a small interaction J coupling the sites and expand ZðtÞ up to second order in J (the first order expansion shows no dependence of the relaxation rate on different temperatures). The covariance of the decoupled conservative system (22)–(25) is more intricate than the covariance of the nonconservative model (11), but the computation is still possible. Performing the integrations and the time Fourier transform on t0 ¼ t1 t2 , we get T uN T v S~ 2 ðu; v; p0 Þ ¼ C 0 T v þ C 1 Ju;j JvþN;j T j þ C 2 Ji;uN Ji;v , T iN
(31)
(recall that u N ¼ v) where C 0 depends on M, z, r, p0 and J, but has no temperature dependence; C 1 and C 2 depends on M, z, r and p0 . If we take J T ¼ J, we obtain Ji;uN ¼ Ju;iN and Ji;v ¼ JvþN;i . In (31), the sum over the indices i and j is considered. Hence, using the latter expressions for J and factoring T v , we get Tj Tv 2 ~ S2 ðu; v; p0 Þ ¼ T v C 0 þ ðJu;j Þ C 1 þ C2 , Tv Tj which reminds (15), specially if we have C 1 ¼ C 2 (a huge computation is necessary to determine the exact relation between C 1 and C 2 ). Anyway, the (gradient) temperature dependence is clear, and properties similar to those already described for the previous nonconservative dynamics follow here. It is worth to emphasize, once more time, the reliability of the perturbative analysis: besides the previous comments on the nonconservative systems, we recall that, in the study of heat conduction for a harmonic chain with weak interactions (related to the study of a two-point function in the steady state of these conservative models), a perturbative analysis [17] gives the same result of the complete and rigorous treatment [9].
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7. Concluding remarks In order to search for basic properties of nonequilibrium Langevin systems, we analyze some harmonic models with dissipative and conservative dynamics, and investigate the time relaxation rate of fluctuations in their two-point function. We show that the time relaxation rate depends on the temperature in the case of fluctuations in the steady state of a system submitted to a temperature gradient, a property which does not exist in the case of convergence to equilibrium (i.e., for a system submitted to an unique temperature). In short, we present here a property of nonequilibrium statistical models which is absent from equilibrium systems. And, it is worth to recall, such property is directly related to observable effects in concrete physical systems. Finally, we stress that the perturbative calculation carried out within the integral formalism is not naive, as pointed out by the comparisons between rigorous and perturbative results recalled for the cases of dissipative and conservative dynamics. Acknowledgment Work partially supported by CNPq (Brazil). References [1] D. Ruelle, Nature 414 (2001) 263–264. [2] G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74 (1995) 2694–2697; G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80 (1995) 931–970. [3] B. Derrida, J.L. Lebowitz, E.R. Speer, Phys. Rev. Lett. 87 (2001) 150601. [4] F. Bonetto, J.L. Lebowitz, L. Rey-Bellet, in: A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinsk (Eds.), Mathematical Physics 2000, Imperial College Press, London, 2000, pp. 128–150. [5] S. Lepri, R. Livi, A. Politi, Phys. Rep. 377 (2003) 1–80. [6] O.V. Gendelman, A.V. Savin, Phys. Rev. Lett. 84 (2000) 2381–2384. [7] O.V. Gendelman, A.V. Savin, Phys. Rev. Lett. 94 (2005) 219405. [8] L. Yang, B. Hu, Phys. Rev. Lett. 94 (2005) 219404. [9] F. Bonetto, J.L. Lebowitz, J. Lukkarinen, J. Stat. Phys. 116 (2004) 783–813. [10] A. Dhar, Phys. Rev. Lett. 86 (2001) 3554–3557. [11] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435–479. [12] P.A. Faria da Veiga, M. O’Carroll, E. Pereira, R. Schor, Commun. Math. Phys. 220 (2001) 377–402. [13] R. Schor, J.C.A. Barata, P.A. Faria da Veiga, E. Pereira, Phys. Rev. E 59 (1999) 2689–2694. [14] R.S. Thebaldi, E. Pereira, A. Procacci, J. Math. Phys. 46 (2005) 53302. [15] E. Pereira, Phys. Rev. E 65 (2002) 56605. [16] E. Pereira, Physica D 192 (2004) 23–32. [17] E. Pereira, R. Falcao, Phys. Rev. E 70 (2004) 46105. [18] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 1991. [19] J. Glimm, A. Jaffe, Quantum Physics, Springer, New York, 1987. [20] M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [21] J. Dimock, J. Stat. Phys. 58 (1990) 1181–1207. [22] E. Pereira, Lett. Math. Phys. 64 (2003) 129–135.