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IPS Research Report No. 92-29

arXiv:hep-lat/9212033v1 24 Dec 1992IUHET-241IPS Research Report No. 92-29UALG-PHYS-12A realistic heat bath:theory and application to kink-antikink dynamicsA.

Krasnitz* and Robertus Potting**Physics DepartmentIndiana UniversityBloomington, Indiana 47405and*Interdisciplinary Project Center for SupercomputingEidgen¨ossische Technische Hochschule-ZentrumCH-8092 Zurich, Switzerland**Universidade do AlgarveUnidade de Ciˆencias Exactas e HumanasCampus de Gambelas, 8000 Faro, PortugalWe propose a new method of studying a real-time canonical evolution of field-theoretic systems with boundary coupling to a realistic heat bath. In the free-fieldcase the method is equivalent to an infinite extension of the system beyond theboundary, while in the interacting case the extension of the system is done in linearapproximation.

We use this technique to study kink-antikink dynamics in ϕ4 fieldtheory in 1+1 dimensions.1

I. INTRODUCTIONThere is a growing interest in transitions over energy barriers in field theories at fi-nite temperature.The motivation comes from high-energy physics and cosmology (e.g.,baryon-number violating sphaleron transitions in electroweak theory [1]- [6]), as well asfrom condensed-matter physics (e.g., current-reducing fluctuations in one-dimensional su-perconductors [7]). Such thermally induced transitions usually involve collective excitationslike kink-antikink pairs or sphalerons, and therefore occur in a nonlinear setting.

The non-linearity, in turn, often renders analytic techniques useless, forcing one to resort to numerical(lattice) methods. On the other hand, many questions of interest can be answered by con-sidering the classical regime in which the temperature is much larger than the energies ofthe field quanta at the scale of a relevant collective excitation [6].A fundamental quantity of interest is the corresponding transition rate.

Unlike staticobservables, it cannot be determined from a given canonical ensemble of field configurations.Instead, one must follow a real-time evolution of the field-theoretic system in the heat bath.Obviously, the rate would depend on the properties of the heat bath and its coupling to thesystem. This was indeed confirmed by recent numerical studies [3] [4], in which the heatbath was implemented through Langevin equation.

In most naturally occurring situationsa system interacts with its environment through the boundaries, and the environment is aninfinite extension of the system itself. This consideration dictates our choice of a heat bathin this work.

Ideally, the heat bath we model should possess two main properties: (a) wavestraveling through the boundary out of the system should be completely absorbed, as if therewere no boundary and the system had an infinite extension; and (b) the waves traveling intothe system should be thermally distributed. Our construction of a heat bath ensures thesetwo properties in the free-field case, while for an interacting field they hold in the linearapproximation.

The similarity between our linearized heat bath and the true extension ofthe interacting system becomes better with decreasing temperature. Note that the standardLangevin dynamics, imposed either in the bulk of the system or at the boundary [11], will2

not lead to the properties a and b.While the idea of mimicking a natural heat bath is very generally applicable, we restrictourselves in this article to the case of a scalar field in one spatial dimension. In Section II wederive the boundary force exerted by the heat bath on this one-dimensional system.

Next,we show in Section III how our construction of the realistic heat bath can be implementednumerically. As an example, in Section IV we apply our technique to study kink-antikink pairnucleation in ϕ4 theory, a subject that has been extensively investigated by other methods[4]- [6].

We summarize and discuss our results in Section V.II. THE METHODConsider a lattice system consisting of a (possibly self-interacting) scalar field ϕn, witha minimum of its potential V (ϕ) at ϕ = v and mass m corresponding to that minimum.

Fordefiniteness, let it reside on the n ≤0 sites of a one-dimensional chain, and be coupled to aheat bath at n = 0. Let the heat bath be a free massive scalar field system on the positiven sites with equation of motion (the lattice spacing is a)¨Φn −Φn+1 + Φn−1 −2Φna2+ m2Φn = 0(1)for n ≥1 and boundary conditionΦ0(t) = ϕ0(t) −v = f(t).

(2)Given the solution Φn(t), there will be a reaction force exerted by the heat bath on thesystem:F(t) = (Φ1 −Φ0)/a2. (3)The Φ0 contribution to F(t) is a harmonic force.

The Φ1 contribution, called fmem (thememory force) in the following, is, as we shall see immediately, a response of a more generalnature. As the heat bath is a free field system, this response should be linear.

Furthermore,3

it should causally depend on f(t). In other words, there exists a response function χ(t)(t > 0) such thatfmem(t) =Z t−∞f(t′)χ(t −t′)dt′.

(4)In order to determine χ(t) let us take f(t) = est.It follows from Eq. 1 that Φn(t) =exp(st −ksan), withks = 2a arcsinha2√s2 + m2.

(5)Using Eq. 3 we then find that χ(t) should satisfy1a2est−ksa =Z t−∞est′χ(t −t′)dt′ = est(Lχ)(s),(6)where L denotes the Laplace transform.

It follows thatχ(t) = a−2L−1(e−aks)= 1a2L−1(q1 + y −√y)2with y = a2(s2 + m2)/4. (7)The integral in Eq.

4 is most easily computed in Fourier space.We define ˜ft(ω) =R t−∞eiω(t′−t)f(t′)dt′ and ˜χ(ω) =R ∞−∞eiωtχ(t)dt, where we are free to take any extensionof χ(t) for t < 0. Then it follows from the convolution theorem thatfmem(t) = 12πZ˜χ(ω) ˜ft(ω)dω.

(8)For numerical purposes it is best to extend χ(t) to t < 0 such that ˜χ(ω) vanishes outsidea finite interval. To this end we choose χ(t) = −χ(−t), so ˜χ(ω) becomes purely imaginary(essentially the sine transform).

The latter can be obtained immediately from the Laplacetransform (Eq. 7) by the π/2 rotation of s in the complex plane; taking the imaginary partthen gives˜χ(ω) = isign(ω)q(ω2 −m2)(1 + a2(m2 −ω2)/4)(9)for frequencies m < |ω|

We expect the heat bath response to be finite for any bounded f(t) (Eq. 4).

If so, χ(t)must decay sufficiently rapidly in the distant future. This indeed is the case: using Eq.

9and performing the inverse Fourier transform we find in the saddle-point approximationχ(t) ∝t−3/2 sin(mt + π/4) as t →∞.The dissipation by the heat bath can equivalently be described by a linear response tothe momentum π0 at the boundary, rather than the field. Explicitly, it follows from Eq.

4thatfmem(t) = −Z t−∞ψ(t −t′)π0(t′)dt′,(10)where ψ(t) =R t0 χ(t′)dt′ or, equivalently, (Lψ)(s) = s−1(Lχ)(s), so that˜ψ(ω) = iω−1 ˜χ(ω). (11)We now turn to the second contribution to the boundary force, the random force fran.It is a Gaussian random variable whose properties are defined by its time autocorrelationC(τ) = ⟨fran(t)fran(t + τ)⟩.These are most easily determined through the fluctuation-dissipation theorem, statingC(τ) = θψ(|τ|)(12)(θ denotes the temperature).Numerical implementation of fran amounts to generatingGaussian noise with this autocorrelation function.One can check explicitly that the time correlation function of the total boundary forceF + fran is equal to the average of ((Φn(0) −Φn−1(0))((Φn(t) −Φn−1(t))/a4 for a canonicalensemble, calculated for an infinitely extended free field.

This is to be expected, as (Φn −Φn−1)/a2 represents the force between neighboring sites.We conclude this section by discussing the continuum limit of the heat bath response.Obviously, F(t) diverges as a →0. Instead, aF(t) is a well-behaved quantity whose limitis simply ∂xΦ(x = 0, t), where x is the continuum spatial coordinate.

In other words, if weprescribe the boundary field motion, the heat bath response will determine the boundary5

field spatial derivative in such a way that the waves traveling out of the system will notbe reflected at the boundary. The corresponding response function is found following stepsanalogous to Eq.

4–Eq. 7.

We write down the continuum equations of motion¨Φ −∂2xΦ + m2Φ = 0(x > 0)(13)together with the boundary condition Φ(0, t) = f(t) and require∂xΦ(0, t) =Z t−∞f(t′)χc(t −t′)dt′. (14)Taking f(t) = exp(st) then yields χc(t) = L−1(√s2 + m2), or χc(t) = δ′(t−0+)+mt−1J1(mt).It is an easy exercise to verify that the same result is obtained by first computing the responsefunction for the lattice boundary field derivative and then taking the limit a →0.III.

NUMERICAL IMPLEMENTATIONOur numerical implementation of the boundary force is as follows. The memory forcefmem(t) is computed using Eq.

8, that is, in Fourier space. The function ˜χ(ω) is given byEq.

9. In practice, the integral Eq.

8 is replaced by a finite sum over discrete values of ω,separated by an increment ∆ω. As a result, the response function χ(t) becomes periodicin t with a period 2π/∆ω.

Beyond t = 2π/∆ω our approximation of χ(t) is incorrect, andthe lower limit of integration in Eq. 4 should be cut off.

We achieve this by changing thedefinition of ˜ft(ω) to˜ft(ω) =Z tt−T eiω(t′−t)f(t′)dt′,(15)where T < 2π/∆ω. At the same time, T should be large enough so the discarded part of theintegral Eq.

8 is negligible. With the new definition, ˜ft(ω) satisfies the equation of motion˙˜ft(ω) = −iω ˜ft(ω) + f(t) −e−iωTf(t −T).

(16)The random force fran is computed by convolving white noise of unit power spectrum witha function R(t) whose Fourier image is given byq˜C(ω), where ˜C is the Fourier transformof C(t) given by Eq. 12 [9].

This can be done very efficiently using FFT algorithms.6

With both ingredients of the boundary force in place, we can now write down andintegrate the equations of motion for the one-dimensional system immersed in the heatbath. These equations have the standard form¨ϕn −ϕn+1 + ϕn−1 −2ϕna2−V ′(ϕn) = 0(17)in the bulk of the system, but should be modified at the boundaries.

For example, at theleft boundary we have¨ϕ0 −ϕ1 −2ϕ0a2−V ′(ϕ0) = fmem,left + fran,left,(18)and the right-boundary analog is obvious. The system of equations of motion for the field issupplemented by Eq.

16 governing the evolution of the memory forces at both boundaries.We integrate equations 17, 18, and 16 using the second-order Runge-Kutta algorithm.While this way of updating ˜ft(ω) is efficient compared to computation of the full integralEq. 15, it is not accurate enough to maintain correct phases of ˜ft(ω) for times much longerthan 2π/∆ω.

To ensure stability, we therefore adjust the values of ˜ft(ω) by computingEq. 15 every T time units.We have tested the action of fmem by evolving an initially hot system with fran omitted.This corresponds to cooling in the zero-temperature heat bath.

The resulting evolution isvery similar to that of a benchmark run where we substitute for the heat bath a real coldfree-field system with large volume. This similarity holds separately for every momentummode.

The method allows cooling the system to at least 10−5 times its original temperature.Cooling curves for a system with a real and with a simulated heat bath are shown in Figure1.We then turned fran on and tested it by heating a cold field configuration to a prescribedtemperature. Again it compared well, mode by mode, with a benchmark run, in which wecoupled a cold system to a large real heat bath whose temperature the system eventuallyreached (let us stress that using the real heat bath is much more costly in terms of CPUtime).

Both heating curves are shown in Figure 2.7

Finally, we have shown numerically the self-consistency of our method by comparing themotion of the endpoint field to that in the middle of the system [10]. As Figure 3 shows,the autocorrelation curves of the two fields are very close to each other, meaning that thesimulated free-field heat bath closely approximates the real one.IV.

KINK-ANTIKINK DYNAMICS IN 1+1 DIMENSIONSWe have applied our method to investigate kink-antikink dynamics of ϕ4 theory whoseLagrangian isL =Zdx12˙ϕ2 −(∂xϕ)2−14(ϕ2 −1)2(19)in suitably chosen units [6]. Kinks and antikinks are finite-energy solutions of the equationsof motion interpolating between the vacuum values of the field ϕ = ±1.

Explicit func-tional form of the static kink is ϕ±(x) = tanh(± x√2). The spatial extension of a kink isapproximately√2, and the kink mass is M =q8/9.Following [4], we chose a system of N = 400 sites with lattice spacing a = 0.5.

Thesimulations were performed at seven values of inverse temperature β between 3.0 and 6.0.Our Runge-Kutta time step was 0.005. For each value of the temperature we started froman ordered system and warmed it up for 2500 time units followed by 2 × 105 time units overwhich we measured the number of kinks n. For the latter we used the same definition as in[4]: the number of zeros in a field configuration smoothened over the physical distance of∆L = 5.The average kink number ⟨n⟩at a given temperature should not depend on the propertiesof a heat bath.

Our measurements of this observable, shown in Figure 4, are indeed closeto those of [4]. Note, however, that our measurement errors are much larger, especially forlow temperatures, even though our data sample is as big as in [4].

We estimated the errorsusing a jackknife method with the block size varying over a very long range; we are thereforeconfident that the autocorrelation of our data is properly taken into account. Moreover, we8

studied a microcanonical evolution of our system at the energy roughly corresponding toβ = 4.5, with the error estimate similar to that of the corresponding canonical case.The temperature dependence of ⟨n⟩may be interpreted in terms of the effective kinkmass. Namely, one expects [8]⟨n⟩∝qβ exp(−βMeff).

(20)It was found in earlier work that Meff< M. It was also indicated that Meffis temperaturedependent [4], [5]. Both features find further evidence in our study.

If we try to fit all ourmeasurements of ⟨n⟩to Eq. 20 at once, an unacceptably low goodness-of-fit results.

Thesituation improves dramatically if we exclude the highest-temperature point from the fit.We then find Meff= 0.695 ± 0.0095, or Meff= (0.737 ± 0.010)M, in good agreement with[4]. Alternatively, we can use pairs of consecutive values of ⟨n⟩to extract Meff.

The result,presented in Figure 5, shows the tendency of Meffto decrease at higher temperatures, inagreement with findings of [4], [5].Another interesting quantity we extract from the kink-antikink number time history isa kink lifetime, i.e. the autocorrelation time τ of n. The latter is usually obtained by fittingthe n autocorrelation function ⟨(n(t) −⟨n⟩)(n(0) −⟨n⟩)⟩to a single exponential of the formexp(−t/τ).

If the time history exhibits more than one time scale (as is the case for n), τcan only be given an average, or effective meaning. The existence of such multiple scalesalso makes a single-exponential fit to the autocorrelation function extremely difficult.

Amultiexponential fit to noisy data is not a practical possibility. We use an alternative way ofdetermining τ, closely related to the integral definition of the autocorrelation time.

Namely,if ∆t is a time interval between two consecutive measurements of n, we expectw(N) ≡⟨(N−1Xi=0n(i∆t) −N⟨n⟩)2⟩(21)≈⟨(n −⟨n⟩)2⟩Ntanh∆t2τ −12 sinh2 ∆t2τ.Obviously, for large N w(N) approaches a random-walk behavior. For a given N we de-termine w(N) from our data set and solve Eq.

21 for τ. For t ≡N∆t >> τ the result9

is approximately independent of N, and we take it as an estimate of the kink lifetime. Atypical dependence of τ on t is shown in Figure 6.

Note that large values of t for which theplateau is reached indicate the existence of multiple time scales in the kink-antikink numberfluctuations.Unlike the exponential fit, this method also allows a well-defined error-estimating proce-dure for τ. In particular, we apply the jackknife technique.

Note that the lag values t usedto determine τ are of the order of 5000. This is to be compared to our n time history lengthof 2×105.

We therefore only have a small effective number of independent measurements ofτ, and our error estimate cannot be very accurate. Conservatively we can expect the errorsof τ to be correct within a factor of 2.

This might explain their inhomogeneous dependenceon the temperature.Kink-antikink pair nucleation can be viewed as a multidimensional analog of a particleescape over a barrier [2], [8], with the kink lifetime related to to the effective barrier heightB [4]:τ ∝exp (βU) ,(22)where U = B −Meff. From our data (Figure 7) we find U = 1.01 ± 0.06 = (1.07 ± 0.06)M,slightly higher than U = (0.85 ±0.15)M of [4], obtained by solving a low-viscosity Langevinequation.

Both that value and ours are inconsistent with the naively expected B = 2M.More work is required to explain this discrepancy [12]. More importantly, however, ourresults show no exponential suppression of the kink-antikink pairs nucleation rate withgrowing temperature, in agreement with analytical predictions and earlier numerical work[2]- [6].V.

CONCLUSIONS AND OUTLOOKIn this article we presented a method of modeling naturally occurring heat baths. Whileour presentation concentrated on a scalar field in one spatial dimension, the principles un-derlying our construction of a heat bath do not depend on the dimensionality or the field10

content of a theory in question. In any case, one can determine the memory force exerted bya heat bath by studying the linear response of the latter to the field motion at the systemboundary.

The corresponding random component of the force may then be found usingfluctuation-dissipation theorem. The only new feature to appear in dimensions higher thanone is related to the connectedness of the boundary: the field motion at different pointsof the boundary will be correlated in a way consistent with causality.

This is, however,a technical difficulty, not a conceptual one. The work on extending our method to othersystems is currently in progress.We have verified numerically that our simulated heat bath thermalizes correctly bothlinear and nonlinear systems.

As an application, we considered the dynamics of kink-antikinkpairs in ϕ4 theory. Our measurements of the kink density agree well with those obtainedby solving Langevin equation, as one would expect for an equilibrium quantity independentof a heat bath implementation.

The kink lifetimes we measure are close to those followingfrom the low-viscosity Langevin dynamics. This is again to be expected for sufficiently largesystems: as the system size grows, the influence of the boundary heat bath on the dynamicsdecreases.

The Langevin analog would then be decreasing viscosity.To conclude, we emphasize again an important advantage of our simulated heat bathover the Langevin method (including its zero-viscosity microcanonical limit): our heat bathis not arbitrarily chosen and involves no free parameters like viscosity. Rather, it is, to thebest of our knowledge, the closest known approximation to a natural situation, in whichopen systems are immersed in a similar environment.

The only physical (not numerical)approximation we make is linearization: our heat bath exchanges linear excitations (planewaves, or mesons) with the system, but not nonlinear ones (kinks, or baryons). Since weare only interested in pair creation and annihilation processes, the exchange of topologicalcharge with the environment should have little impact on our results.

It follows that we canapproximately (the only approximation being linearization of the heat bath) identify oursimulation time with the real physical one. The transition rates we measure have thereforedirect physical meaning.11

ACKNOWLEDGMENTSWe are indebted to Ph. de Forcrand, to S. Gottlieb, and to A. Kovner for many interestingdiscussions and helpful suggestions.

This work was supported by the US Department ofEnergy and by the Swiss Nationalfond.12

REFERENCES1For a review see, e.g., A. D. Dolgov, Phys. Rep. 222, 309 (1992).2M.

Dine, O. Lechtenfeld, B. Sakita, W. Fischler, and J. Polchinski, Nucl. Phys.

B 342,381 (1990).3A. I. Bochkarev and Ph.

de Forcrand, ”Nonperturbative evaluation of the diffusion ratein field theory at high temperatures”, IPS Research Report No. 92-18, October 1992.4M.

Alford, H. Feldman, and M. Gleiser, Phys. Rev.

Lett. 68, 1645 (1992).5A.

I. Bochkarev and Ph. de Forcrand, Phys.

Rev. Lett.

63, 2337 (1989).6D. Grigoriev and V. Rubakov, Nucl.

Phys. B 299, 67 (1988).7N.

Giordano, Phys. Rev.

B 41, 6350 (1990) and references therein.8P. H¨anggi, F. Marchesoni, and P. Sodano, Phys.

Rev. Lett.

60, 2563 (1988); F. March-esoni, ibid 64, 2212 (1990).9In fact, the choice of phase for the Fourier transform of R(t) is arbitrary. There is aunique choice which would make R(t) causal.

In the present work we choose zero phasefor convenience.10We thank Ph. de Forcrand for suggesting this test.11Z.

Rieder, J. L. Lebowitz, and E. Lieb, J. Math.

Phys. 8, 1073 (1967); U. Z¨urcher andP.

Talkner, Phys. Rev.

A 42, 3267 (1990); ibid, 3278 (1990).12A word of caution is due regarding any attempt at numerical verification of analyticalpredictions for kink-antikink dynamics. Namely, both static and dynamical propertiesof the kink-antikink gas are sensitive to the way the kink-antikink number is defined, i.e.to the smearing length ∆L.

To check the ∆L dependence of our results we determined⟨n⟩and τ at β = 3.0, 4.5, and 6.0 for a range of ∆L values between 0.5 and 5. Both Meffand U evidently are ∆L-dependent.

For example, U varies from 0.66 to 1.01 for ∆L in13

that range. On the other hand, the analytical predictions do not take ∆L into accountat all, and can therefore only serve as a qualitative guide for interpreting numericalexperiments.14

FIGURESFIG. 1.

Cooling curves for a free-field system (m = 0.5, a = 1, L = 100) immersed in a simulated(squares) and real (solid line) zero-temperature heat baths.FIG. 2.

Heating curves for a free-field system (m = 0.5, a = 1, L = 100) immersed in a simulated(squares) and real (solid line) heat baths. The equilibrium is reached at θ = 1.FIG.

3. Autocorrelation functions of the endpoint (solid line) and midpoint (squares) fields fora free-field system (m = 0.5, a = 1, L = 100) at θ = 1.FIG.

4. Temperature dependence of the kink number (logarithmic scale).

The solid line is a fitfor Meff= 0.737M.FIG. 5.

Temperature dependence of the kink effective mass obtained by fitting pairs of consec-utive points from Figure 4 to Eq. 20.

The inverse temperature β is that of a higher-temperaturepoint in each pair.FIG. 6.

The kink number autocorrelation time τ at β = 4 determined from Eq. 21, plotted asa function of the lag N∆t.FIG.

7. Temperature dependence of the kink lifetime (logarithmic scale).

The solid line is a fitfor U = 1.06M.15

FIGURE 116

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FIGURE 520

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FIGURE 722

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