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NONPERTURBATIVE EVALUATION OF THE DIFFUSION

arXiv:hep-lat/9210027v1 22 Oct 1992NONPERTURBATIVE EVALUATION OF THE DIFFUSIONRATE IN FIELD THEORY AT HIGH TEMPERATURESAlexander Bochkarev a∗† and Philippe de Forcrand baInstitute for Theoretical Physics, University of BernSidlerstrasse 5, CH-3012 Bern, Switzerland.bIPS, ETH Zurich, CH-8092 Zurich, SwitzerlandAbstractKramer’s approach to the rate of the thermally activated escape froma metastable state is extended to field theory.Diffusion rate in the 1+1-dimensional Sine-Gordon model as a function of temperature and friction coef-ficient is evaluated numerically by solving the Langevin equation in real time.A clear crossover from the semiclassical to the high temperature domain is ob-served. The temperature behaviour of the diffusion rate allows one to determinethe kink mass which is found equal to the corresponding classical value.

TheKramer’s predictions for the dependence on viscosity are qualitatively valid inthis multidimensional case. In the limit of vanishing friction the diffusion rateis shown to coincide with the one obtained from the direct measurements ofthe conventional classical real-time Green function at finite temperature.∗On leave of absence from INR, Russian Academy of Science, Moscow 117312, Russia†Work supported by Tomalla-Stiftung

1IntroductionStatistical systems with finite energy barriers separating different domains of thephase space may exhibit metastable states. Evaluation of the rate of escape fromthose states is an important dynamical problem.

The simple example of a statisticalsystem where one degree of freedom is exposed to the potential illustrated by Fig.1was first considered by Kramers [1]. At temperatures :ω0 ≪T ≪ES(1.1)where ω0 is the scale of quantum fluctuations and ES is the barrier height, the classicalthermodynamical fluctuations contribute to the escape rate Γ ∼exp(−ES/T) of thesystem initially localized in the well.

Kramers [1] suggested to determine the escaperate by means of the classical Langevin equation:˙φ = p / M ,˙p = −∂U∂φ −γp + η(t)(1.2)⟨η(t) η(0)⟩= 2TγMδ(t)(1.3)where γ is a friction coefficient, introduced as an input parameter in this phenomeno-logical description and M corresponds to the mass of the would be Brownian parti-cle. γ determines how strong is the coupling of the system to the heat-bath, repre-sented by the white noise η , and therefore how fast the system reaches equilibrium.The normalization of the random force (1.3) (the Einstein relation) comes from thefluctuation-dissipation theorem.In the case of one degree of freedom the escaperate Γ(T, γ) has been obtained analytically [2] in the domain of very small γ andfor moderate-to-large friction γ ≫ω−, where ω−is the negative eigenmode of thefluctuations near the stationary point φ = φs.

In quantum field theory one oftendeals with an analogous problem of penetration through some energy barrier. Onevery important example is anomalous fermion (or axial charge) number violationin gauge theories with nontrivial structure of the ground state [3].

Here the staticenergy barrier separates different classical vacua with definite integer Chern-Simons1

numbers [4]. Penetration through this energy barrier leads to the dissipation of thefermionic number.

The system at high temperature is expected to exhibit thermalactivation behaviour of the rate of anomalous fermion number non-conservation [5].The problem specific for the field theory is that the energy barrier - analogue of Uin Eq. (1.2) - is not known explicitly.

This energy barrier is in the multidimensionalconfigurational space. Only the static field configuration, corresponding to the top ofthat barrier is known from topological considerations [6] or explicitly in some models[7].In previous works [8] , [9] we suggested to follow Kramers in studying the energybarrier in field theory.

In [9] we introduced microscopical classical Langevin equationin real-time in the 1+1 dimensional Abelian-Higgs theory and observed the Brownianmotion of the Chern-Simons variable between topologically distinct classical vacua.Our measurements of the diffusion rate revealed its thermal activation dependenceon the temperature in the domain (1.1). Although this approach is applicable in theinteresting domain of high temperatures, where the semiclassical approximation isnot valid, it was difficult to get data there, because it required rather big lattices.Meanwhile, it was argued in [10] that our results were difficult to interpret, since theywere based on the first-order Langevin equation, derived normally for large frictioncoefficients, while, in fact, γ = 1 had been used.In the present paper we deal with a simpler model, the Sine-Gordon field theoryin 1+1 dimensions, which allows one to obtain not only accurate quantitative resultsbut also to check qualitative conclusions valid beyond this particular model.

In fact,this model is very similar to the Abelian Higgs theory. On the classical level the Sine-Gordon theory has an infinite number of degenerate vacua with finite energy barrierseparating them.

At nonzero temperatures one observes a random walk between thosevacua. At temperatures (1.1) this random process goes through the formation of kink-antikink pairs, so the rate of the process (diffusion rate) is sensitive to the densityof kinks.

Previous studies of the density of kink gas as a function of temperaturein the 1 + 1 dimensional scalar field theory with the double well potential revealed2

an intriguing fact : the thermal activation behaviour of the density of kinks n ∼exp(−EeffK /T) involved an effective kink mass EeffKwhich was found to be 20 ÷ 30%smaller than the classical value EclK [8], [10]. One source of uncertainties in measuringthe density of kinks was the adhoc criterion used in counting the number of kinks in agiven field configuration.

This criterion was suggested in [13]. One advantage of theSine-Gordon model is the possibility to avoid this criterion problem.

The quantitywe measure is very well defined: it is the diffusion parameter of the average field.We measure the diffusion rate in the semiclassical domain (1.1) and beyond. Themeasurements allow us to extract the value of the kink mass EeffK , which we find tocoincide with the classical kink mass EclK.

We therefore conclude that the previouslyobserved discrepancy was an artifact of the criterion used.We also measure the viscosity dependence of the rate by means of the second orderLangevin equation. We find a remarkable coincidence of this dependence with theone predicted by the first order Langevin equation down to very small values of thefriction coefficient.

This implies that one can use the first order Langevin approachin field theory, in particular, for γ ∼O(1).In the limit of vanishing friction we find a nonzero diffusion rate. Since in thislimit the Langevin equations (1.2) become the Hamiltonian ones, we perform a directnumerical measurement of the classical real-time two-point Green function, whichdescribes the diffusion, as a Gibbs’s average.

These ensemble averages give the samethermal activation behaviour of the rate as obtained in the Langevin measurementsand, what is less trivial, they prove to be equal in magnitude to the Langevin mea-surements in the limit of vanishing friction. This establishes an important equivalencebetween the Gibbs ensemble measurements of the real time classical Green functionsat finite temperature and the corresponding Langevin measurements done in theasymptotic domain of very small friction.In the next section we explain the relevance of simulations of the classical fieldtheories to the behaviour of quantum field theories at high temperatures.

We empha-size a particular relation between the lattice spacing of the classical systems under3

consideration and the temperature. In section 3 we briefly introduce the 1 + 1 di-mensional Sine-Gordon model at finite temperature.Then we describe the first-and second-order Langevin equations.

Sections 4 and 5 contain the results of ournumerical simulations.2Classical systems - for quantum theoriesIt is well known that the partition function of a D+1-dimensional quantum field the-ory in the limit of high temperature may be obtained by means of the correspondingD-dimensional classical field theory. To see how this emerges consider, for instance,a scalar field theory in D + 1 - dimensions, defined by the action of the general form:S =ZdD+1x 12(∂φ)2 + m22 φ2 + λUint!

(2.1)The partition function of the corresponding statistical system is given by functionalintegral with the following Matsubara action:SM =Z β0 dx4ZdDx 12(1¯h∂4φ)2 + 12(∇φ)2 + m22 φ2 + λUint! (2.2)where β = 1/T is inverse temperature, ¯h is Planck constant and periodic boundaryconditions are imposed on the field in the imaginary time direction.

Eq. (2.2) impliesthat in any of the limits :¯h →0 , or T →∞(2.3)the static x4-independent field configurations dominate in the functional integral.

Ifone ignores for a moment ultraviolet divergences and performs the limit (2.3) formally,the contribution of the static modes is determined by the actionSeff = βZdDx 12(∇φ)2 + m22 φ2 + λUint! (2.4)This action does not mention Planck constant.The functional integral with theaction (2.4) determines the partition function of the corresponding classical system.To be more accurate, one should keep in mind that nonstatic modes do not decouple4

in the ultraviolet divergent diagrams, which means that the action (2.4) involvesrenormalized Largangian parameters - running coupling constants, normalized onthe temperature. The renormalization effects are the memory about the quantumnature of the original theory (2.2).

To obtain the effective action (2.4) is, in general,a separate problem [11]. In some simple cases in low dimensions the action (2.4) issimply a static Hamiltonian.Let us see explicitly how the reduction takes place, for example, in the case of weakinteraction (λ ≪1).

The free energy of a gas of particles of mass m, correspondingto (2.1), in the one-loop approximation reads as:F = TVXnZdkD(2π)D12 logω2n + ω2k(2.5)where ωn = 2πnT, n = 0, ±1, ±2, ... are Matsubara frequencies, ωk =√m2 + k2 andV is D-dimensional volume. The temperature dependent partF = TVZdkD(2π)D log (1 −exp (−βωk))(2.6)becomes in the limit T ≫m :F = −TVZdkD(2π)D βknB(βk)(2.7)where nB is Bose distribution so thatF = −κ V T D+1(2.8)withκ =Ω(2π)D Γ(D + 1)ζ(D + 1)(2.9)where Ω= 2πD/2/Γ(D/2) is a surface of the D - dimensional sphere [12].If we want to obtain the same expression by means of the effective theory (2.4) wehave to perform the limit β →0 in the integrand of Eq.

(2.7), which gives divergentfactor:F = −TVZdkD(2π)D(2.10)5

This divergent integral counts the number of degrees of freedom of the classical fieldtheory (the Rayleigh-Jeans divergency). Thus the partition function of the classicalfield theory is something which needs to be defined.

Regularization is necessary forthis end.The vacuum energy of the effective theory (2.4).F = −TVZdkD(2π)D12 log GΛ(2.11)is determined by the regularised propagatorGΛ = 1/ω2k + ω4k/Λ2(2.12)For the temperature dependent cut-off:Λ = cT , c ∝O(1)(2.13)one recovers the result Eq. (2.8) with c determined by κ.This simple exercise illustrates one general and important statement.

The effectiveD - dimensional classical field theory (2.4), which serves for the calculation of the hightemperature limit of the corresponding D+1 - dimensional quantum field theory, has aphysical cut-off, given by Eq.(2.13). If we assume lattice regularization for the originaltheory (2.2), we conclude that the continuum limit is performed simultaneously withthe limit (2.3).

One should perform thermodynamical limit in the functional integralwith the effective action(2.4), but not the continuum one. It is essential that thelattice spacing a−1 ∝Λ here is the physical cut-off, unambiguously fixed by thetemperature:a−1 ∝T /¯h(2.14)In this sense the original quantum field theory may be studied at high temperaturesby simulating the classical functional integral with the regularized action (2.4) anda temperature dependent cutoff(2.14) [13].As soon as nontrivial lattice spacingdependence is observed in a study of the classical systems at high temperatures therelation (2.14) is to be taken into account.6

3Kramers approach in field theoryWe consider the Sine-Gordon model in 1 + 1 dimensions defined by the action:S =Zd2x 12(∂˜φ)2 −m2λ (1 −cos[√λ˜φ(x)])! (3.1)On the classical level the theory has an infinite number of degenerate vacua withφ = 2πn, n = 0, ±1, ±2, .. .

In the quantum theory the mass m determines the scaleof quantum fluctuations and the self-coupling constant λ is bounded [14]: λ < 8π.The well-known static solution to the classical equations of motion interpolatingbetween different classical vacua ˜φk =4√λ arctan (emx) has energyEk = 8m/λ(3.2)which is parametrically large, in the sense that λ is a parameter of the semiclassical(loop) expansion. If m is chosen as a unit of measure, the Lagrangian in Eq.

(3.1)may be rescaled to the one which has no free parameters:S = 1λZd2x12(∂φ)2 −(1 −cos[φ(x)])(3.3)The Sine-Gordon model provides one with a good opportunity to study thermalactivation phenomena in field theory. The potential energy is completely bounded atinfinity for the zero mode ¯ϕ =1LR L0 dxϕ(x), which allows for the Brownian motionof ¯ϕ at finite temperature.

Any initial state localized around one classical vacuum isfar from equilibrium. In the case of periodic boundary conditions fluctuations of thefield between different vacua proceed via formation of a pair of kink and antikink andtheir separation from each other.

Creation and annihilation of kink-antikink pairsis essentially a random walk between different classical vacua. Thus the diffusionrate of ¯ϕ is sensitive to the kink mass, which determines the height of the staticenergy barrier between different vacua.

For periodic boundary conditions kinks andantikinks appear in pairs, so one expects in the semiclassical domain of temperaturesm ≪T ≪Ek :ΓP BC ∼exp( −2Ek/T)(3.4)7

In the case of free boundary conditions a configuration with a single kink mayappear as a result of the thermal fluctuations.The kink’s motion through spacebrings the system from the neighborhood of one classical vacuum to the vicinity ofanother vacuum. Therefore for free boundary conditions we expect:ΓF BC ∼exp( −Ek/T)(3.5)Following Kramers, we describe evolution of some initial non-equilibrium statelocalized around one classical vacuum by solving the real time Langevin equations.Let us discretize the system of size L = Na , where a is lattice spacing.

TheHamiltonian, corresponding to Eq. (3.3) is :˜H =˜K + ˜U(3.6)˜K =NXn=112λaD (˜pn)2(3.7)˜U =NXn=1aDλ 12 (φn+1 −φn)2 /a2 + 1 −cos[φn](3.8)where D is dimensionality of space, D = 1 in our case.

To generate a system with thedensity matrix ˆρ = exp−˜H/T, one naturally employs the second order Langevinequations of the form:˙φn =λaD ˜pn(i)(3.9)˙˜pn = aDλn(φn−1 + φn+1 −2φn) /a2 −sin (φn)o−γ˜pn + ´ηn(3.10)which is a straightforward generalization of Eqs. (1.2), (1.3) to the case of manydegrees of freedom.

Eqs. (3.6)-(3.10) indicate that formally we deal with a classicalgas of particles of mass:M = aD/λ(3.11)from which one concludes that the normalization of the random force, fixed by theEinstein relation (see Eq.

(1.3)), is : ⟨´ηn(t) ´ηm(0)⟩= 2TγaDδ(t)δnm/λ . The averagekinetic energy is independent of the effective mass M according to the usual formulaeof quantum statistics < ˜K > = NT/2.8

The form of the rescaled action Eq. (3.3) implies that the classical dynamics doesnot depend on λ.

To see it explicitly rescale the momentum: ˜p = λp. Then thedensity matrix looks like :ˆρ = exp (−Heff/θeff)(3.12)θeff = λ TaD(3.13)Heff =NXn=1 12(pn)2 + 12 (φn+1 −φn)2 /a2 + 1 −cos[φn](3.14)The corresponding Langevin equations indeed do not contain the parameter of thesemiclassical expansion λ explicitly:(φn(i + 1) −φn(i)) /ε = pn(i)(3.15)(pn(i + 1) −pn(i)) /ε=(φn−1 + φn+1 −2φn) /a2−sin(φn) −γpn + ηn(i)(3.16)where :⟨ηn(i) ηm(j)⟩= 2γθeffεδnmδij(3.17)and we have discretized Langevin time by the amount of ε.

Physical quantities areobtained in the limit ε →0. Notice that the normalization of the white noise (3.17)is different from that in [10].Eqns.

(3.12)-(3.14) define the effective classical system we are going to simulate. Itincorporates the lattice spacing a as an input parameter, determined in the underlyingquantum theory by Eq.(2.14).

The kinetic energy of the effective system (3.12)-(3.14)averaged over the evolution (3.16) may be evaluated to check the effective temperatureθeff:⟨Keff⟩= 12 N θeff(3.18)One may, of course, consider the first-order Langevin equation instead of thesecond-order one. In the Lagrangian formalism, both sets of equations (3.6)-(3.10)9

and (3.12)-(3.14) are equivalent to the following second-order Langevin equation forthe field φ:¨φn + γ ˙φn = −∂Heff∂φn+ ηn(3.19)The second term in the l.h.s. of Eqn.

(3.19) is the damping force. It dominates overthe first one at large friction or large times t > γ−1 .

Then neglecting the first termin the l.h.s. of Eq.

(3.19) one obtains the first order Langevin equation:φn(i + 1) −φn(i) = −εγ∂Heff∂φn+s2 θ εγ aD ξn(3.20)where θ = λT and ξ is a random variable with Gauss’s distribution of variance 1. θ isthe only parameter of the classical model (apart from the lattice spacing). To fix thephysical temperature one needs to know λ.

Since λ is a parameter of the semiclassicalexpansion according to Eqn. (3.3), it is fixed only in the underlying quantum theory.The r.h.s of Eqn.

(3.20) depends on the ratio ε/γ, not separately on ε and γ.This implies a simple friction dependence of, say, the diffusion rate determined bymeans of the first-order Langevin equation: the diffusion rate must be inverselyproportional to the friction coefficient Γ(T, γ) ∼1/γ. Thus within the first-orderLangevin treatment the absolute value of the diffusion rate is fixed in a simple way bythe friction coefficient γ, which is a dimensional quantity.

The second-order Langevinequation, valid for arbitrary friction, predicts nontrivial friction dependence, whichhas been evaluated explicitly by Kramers [1] in the case of one degree of freedomcoupled to the heat bath. In the limit of large γ that friction dependence of theescape rate, of course, reduces to the one we just derived from the first-order Langevinequation.

The question we would like to clarify is how large the friction coefficientmust be in the case of field theory to allow one to use the first-order Langevin equationinstead of the second-order one. Our numerical data proves to be helpful for this end.10

4Numerical simulationsWe have solved the second-order Langevin equations (3.15-3.17) numerically for thesystem of size L = 50, a = 1. This volume is sufficient to accomodate a few kinks(which size is 1 in our units) at low temperatures.

The diffusion parameter ∆(t) hasbeen measured, defined following [9] as:⟨∆(t)⟩= 1toZ to0dt′ { ¯ϕ(t′ + t) −¯ϕ(t′) }2(4.1)The system was observed over 5 · 107 to 2 · 108 iterations (i.e. a time to ∼106).The longer runs were necessary to reduce statistical errors at the lower temperatureswhere the diffusion rate is smaller.

The time step was typically ε = 0.02 . TheBrownian motion of ¯ϕ has been unambiguously observed.

Fig.2 demonstrates thediffusion at temperature T = 6 for 3 different values of the lattice spacing. The rateis obviously insensitive to the lattice spacing and the relatively large value a = 1 maybe used to obtain physical results.

Large values of a are preferable to save computertime.Periodic as well as free boundary conditions were studied.In both cases wemeasured the diffusion rate in the range of temperatures θ ∼= 1.33 ÷ 18.Thesemiclassical domain corresponds to θ ≪8 in accordance with (3.2). Fig.3 showsthe temperature dependence of the diffusion rate on a logarithmic scale.

In the caseof free boundary conditions all the data points starting from T ∼= 3 downwards lieon a straight line.This implies the expected thermal activation behaviour (3.5).The slope of the straight line is seen to be 8, which is exactly the classical valueof the kink mass (3.2). In the case of periodic boundary conditions, a somewhatmore interesting phenomenon is observed: starting around T = 2 the slope of thestraight line changes from 8 to 16 , which corresponds to the classical energy ofa kink-antikink pair.This is again in accordance with the expectations (3.4) forperiodic boundary conditions.

The sensitivity to the boundary conditions appears inthe lowest temperature domain. Fig.4 shows that at higher temperatures there is nodependence on the boundary conditions.

The cross-over temperature below which11

the kink-antikink pair energy is observed in (3.4) depends on the system size L. Thelarger the volume L the smaller that crossover temperature.The diffusion rate can also be measured in the high temperature domain θ ≥8 ,where the semiclassical approximation is not valid. Our numerical results are shownon Fig.5 for both types of boundary conditions.

The diffusion rate does not dependon the choice of boundary conditions as it is not associated with the production ofkink-antikink pairs any more.At high temperatures the Langevin equation (3.20) can be integrated explicitly,because the noise term, whose contribution is proportional to√T, dominates overthe regular force. Then the analytical predition for the rate is :Γ = 2 Tγ L(4.2)The numerical measurements are in perfect agreement with Eqn.

(4.2).The crossover temperature from the semiclassical to the high temperature domainis seen to be around T = 3. Perhaps surprisingly it is less than the kink energy (3.2).To determine it more accurately we have measured the probability distribution of theaveraged field ¯ϕ, whose logarithm determines the effective potential V ( ¯ϕ, T).

Onecan see on Fig.6 that at T = 2 there is a deep parabolic well at the origin, so thatthe field most of the time oscillates around ¯ϕ = 0. Beyond the parabolic well thepotential is flat.

This implies the free motion between the different vacuum sectors,which proceeds via the formation of kink-antikink pairs, seen as the modulation ofthe high frequency oscillations corresponding to the parabolic well. The deeper thewell the better is the very notion of the kink configuration at T ̸= 0.

At temperatureT = 3 the parabolic well is rather shallow, so the low frequency modulations usuallyinterpreted as kinks become hard to identify. The effective potential is almost entirelyflat, which means that the transitions between different classical vacua do not requiresmooth low frequency configurations any more.

This is obviously the boundary ofthe semiclassical domain of temperatures. At temperature T = 6 there is not anysign of the confining parabolic well.12

Thus we have found that the effective potential V ( ¯ϕ, T) of the diffusing variable isflat in the high temperature domain, starting from T ≃3. This result is interestingin the context of the baryogenesis within the standard electroweak theory [16].

In theStandard Model there is a variable, which is expected to diffuse at finite temperature.It is the Chern-Simons variable NCS. Diffusion of NCS leads to the dissipation of thebaryonic number.

This effect could be responsible for the production of the baryonasymmetry in the expanding Universe. In [17] a scenario for baryogenesis within theStandard Model was suggested, in which the crucial assumption was the flat effec-tive potential V (NCS) in the high temperature domain.

While to measure properlyV (NCS) in the Electroweak Theory is not easy, our data in the Sine-Gordon modelindicates that the flat effective potential V (NCS) is a rather plausible assumption.We would like to emphasize again the similarity between the Sine-Gordon model andAbelian Higgs theory in 1+1 - dimensions. The zero mode ¯ϕ of the Sine-Gordontheory plays the role of the Chern-Simons variable in the Abelian Higgs model.

Onecan decompose ϕ →¯ϕ + δϕ, to demonstrate that the effective potential V ( ¯ϕ, T) isa bounded periodic function, allowing for the Brownian motion.We now turn to the dependence on the friction coefficient. The friction coefficientγ is a dimensional input parameter in the Langevin equation (3.19).

It determinesthe scale as it is seen, for instance, from formula (4.2), derived for large γ. Althoughthe friction dependence of the escape rate has been obtained analytically in [1] inthe case of one degree of freedom exposed to the heat bath, it is a less trivial taskin the field theory. We have measured the friction dependence of the diffusion ratenumerically by solving equations (3.15) - (3.16) for the two temperatures T = 2, 6and for various friction coefficients.

The results are shown in Fig. 7, 8.

In boththe semiclassical and high temperature domains we find the same behaviour of therate. The inverse rate varies linearly with the friction coefficient from large γ > 1down to very small γ ∼10−2.

This simple friction dependence coincides with theone predicted by the first order Langevin equation. Therefore we obtain experimentalevidence that the first order Langevin equation may be used to explore the large time13

behaviour of the correlators not only for large, but also for small friction coefficients,in particular, for γ ∼1.One can see from Fig. 8 that the rate does not diverge in the limit γ →0.

Itapproaches some finite asymptotic value. Meanwhile the original Langevin equationsbecome the conventional Hamiltonian ones in the limit γ →0.

Therefore we expectthe rate extracted from the second order Langevin simulations in the limit of vanishingfriction to coincide with the one obtained by means of microcanonical simulations.To verify this conjecture we first of all notice that the microcanonical simulationsof classical systems in real time [13] naturally correspond to some fixed energy, nottemperature. To obtain the time dependent Gibbs averages, one has to average theresult of the microcanonical measurement over the initial field configuration with theGibbs weight:⟨∆(t)⟩=R Dϕ0Dp0 exp ( −H(ϕ0, p0) / T ) { ¯ϕ( t, ϕ0 , p0 ) −¯ϕ(0) }2R Dϕ0Dp0 exp ( −H(ϕ0, p0) / T )(4.3)The problem here is that the value of the diffusion parameter ∆at time t, ob-tained as a result of the Hamiltonian evolution, is a functional of the initial fieldconfiguration {ϕ0, p0} = {ϕ(t = 0), p(t = 0)}.

This functional is not known ex-plicitly. However, following the suggestion by J.Smit [18] we evaluate this functionalnumerically.

This means that to measure the Gibbs averaged diffusion parameterdirectly we calculate numerically (ϕ(t) −ϕ(0))2 from the Hamiltonian evolution fora given starting configuration {ϕ0, p0}, then update this starting configuration bymeans of the Metropolis procedure, corresponding to some temperature T, then domicrocanonical evolution again and so perform the Gibbs average.In this way the diffusion parameter (4.3) has been measured. The function ⟨∆(t)⟩is shown on Fig.

9. One can see a crossover between two domains of time.

At shorttimes t ≤O(102) the deterministic behaviour is observed ⟨∆(t)⟩∼t2, while att ≥O(102) the Brownian motion sets in ⟨∆(t)⟩∼t. The corresponding diffusionrate as a function of temperature is shown on Fig.10.The thermal activationbehaviour is clearly seen at low temperatures.

As a result we observe a remarkable14

coincidence between the data from the second order Langevin simulations in the limitof vanishing friction and the direct Gibbs averages. One obtains Γ( T = 2 ) ≃.3 andΓ( T = 6 ) ≃8.

from both Fig. 8 and Fig.

10. This result answers the question ofhow to relate the Langevin measurements to the direct calculations of the Gibbs’saverage (4.3).5Comments on the errorsAs mentioned above all the runs which results are presented were long enough tomake statistical errors negligible.

The two main sources of systematic errors are thefinite lattice spacing a and the time step ε. The insensitivity to the lattice spacingwas discussed before (see Fig.

2). Here we would like to address the artifacts of thediscretization of the Langevin time.It is known [19] that the finite time step in Langevin simulations makes the tem-perature of the simulated system Teff larger than the input one T: ∆T ≡Teff −T ∼O(ε).

To check the effective temperature we measure the diffusion parameter at rathersmall times, as shown on Fig. 11.

The Brownian diffusion is clearly observed at shorttimes, but it has nothing to do with the motion of the system between topologicallydifferent vacua. At short times the white noise always dominates over the regularforce in the Langevin equation.

Therefore the short time behaviour of ⟨∆(t)⟩is, infact, controlled by the free Langevin equation, which immediately implies diffusionof ¯ϕ. This short time diffusion is already sensitive to the discretization effects.

Itsdiffusion rate is given by Teff, which has been measured numerically and is shown onFig. 12.

One can see that the effective temperature of the simulated system followsthe input one with fairly high accuracy. Since these measurements are obtained fromvery short runs, the computer time needed is quite small.

So we have also checkedthat ∆T = cε, with c = 1.5 ± .5.Another alternative to check the effective temperature of the simulated system isto measure the Langevin average of the canonical momentum squared over the whole15

run: < p2 > = Teff. This way is not easily generalizable for theories where there isa coupling between the canonical momenta and coordinates in the Hamiltonian, likein gauge theories.6ConclusionsThe study of the Sine-Gordon field theory in 1+1-dimensions shows that the classicalLangevin simulations in real time prove to be efficient in obtaining valuable informa-tion about the nonperturbative effects in field theory at high temperatures.

In thesemiclassical domain of temperatures kinks are seen as smooth modulations of thehigh frequency oscillations of the field. Diffusion between different classical vacua isdue to random process of production and free motion of kink-antikink pairs duringthe Langevin evolution.

This is very well confirmed by the measured temperaturedependence of the diffusion rate, which exhibits thermal activation behaviour withthe classical value of the kink mass in the Boltzmann exponent.Crossover is observed between the semiclassical and high-temperature domain,where the semiclassical approximation is not valid. The effective potential of thediffusing variable is found flat at high temperatures.

In the high temperature domainat moderate friction the diffusion rate follows prediction of the free Langevin equation.The dependence of the diffusion rate on the friction coefficient is found identicalto the prediction of the first order Langevin equation down to rather small frictioncoefficients γ ∼10−2. This justifies the use of the first order Langevin equation inthe calculations of correlation functions at large real times.We also measure the diffusion parameter directly as the Gibbs ensemble average bymeans of the Metropolis procedure and microcanonical evolution.

The correspond-ing diffusion rate is found to coincide with the smooth extrapolation of Langevinmeasurements in the limit of vanishing friction coefficient.16

7AcknowledgementsFruitful discussions with P. van Baal, O. Lanford, H. Leutwyler, P. Hasenfratz, J.Hetrick, J. Smit are gratefully acknowledged.17

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Figure CaptionsFig. 1 Typical potential creating a metastable state at finite temperature.Fig.

2 Time dependence of the diffusion parameter (4.1) obtained from the secondorder Langevin equation. Measurements were performed every 100 iterations.

Theparameters are: T = 6, L = 50, ε = .02. Three solid lines correspond to differentlattice spacings: a = {1, .5, .25} ; the dashed line is a fit with slope 1.Fig.

3 Temperature dependence of the diffusion rate on a logarithmic scale for:a) free, b) periodic boundary conditions. L = 50, a = 1.

Typical uncertainty isshown for T = 2. The dashed lines correspond to the Boltzmann exponent withslope given by the classical value of the kink mass Mk = 8 in a) and the energy ofthe kink-antikink pair 2Mk = 16 in b).Fig.

4 Temperature dependence for free vs. periodic boundary conditions at bothhigh and low temperatures.Fig. 5 Temperature dependence of the diffusion rate at high temperatures.

Thedashed line is a prediction from the corresponding first order free Langevin equation.Fig.6 Logarithm of the probability distribution (effective potential) of ¯ϕ atdifferent temperatures indicated on the plots.Fig. 7 Friction dependence of the rate in the low (a) and high (b) temperaturedomains obtained from the second order Langevin equation.

L = 50, a = 1. Thedashed straight line is a prediction from the first order Langevin equation, derivednormally for large friction γ > 1 or large times.Fig.

8 Same as in Fig. 7, but only the points corresponding to very small valuesof the friction coefficient are shown.

The cross-points on the y-axis are the extrapo-lations.Fig. 9 Time dependence of the diffusion parameter (4.1) obtained from direct mea-surements of the Gibbs average (4.3) for periodic boundary conditions.

Temperatureis T = 1.5. The initial field configuration was updated by means of the Metropo-lis procedure 100 times and each time 106 leap-frog iterations of the microcanonical20

evolution were performed. L = 50, a = 1, ε = .1Fig.

10 Temperature dependence of the diffusion rate obtained from direct mea-surements of the Gibbs average (4.3). The dashed line corresponds to the Boltzmannfactor with the energy of the kink-antikink pair.

The parameters of the measurementsare as in Fig. 9.Fig.11 The typical diffusion at short times, controlled by the free Langevinequation.Fig.12 The effective temperature of the simulated system obtained from theshort time diffusion.

The dashed line is a fit. The solid line corresponds to the inputtemperature.

The deviation is proportional to the time step ε.21

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