Functional Renormalisation Group for Brownian Motion II: Accelerated Dynamics in and out of Equilibrium
FFunctional Renormalisation Group for Brownian Motion II:Accelerated Dynamics in and out of Equilibrium
Ashley Wilkins ∗ and Gerasimos Rigopoulos † School of Mathematics, Statistics and Physics, Newcastle University,Newcastle upon Tyne, NE1 7RU, United Kingdom
Enrico Masoero ‡ School of Engineering, Newcastle University,Newcastle upon Tyne, NE1 7RU, United Kingdom (Dated: February 10, 2021)Here we numerically solve the equations derived in part I of this two-part series and verify theirvalidity. In particular we use the functional Renormalisation Group (fRG) flow equations to ob-tain effective potentials for initially highly anharmonic and non-polynomial potentials, includingpotentials with multiple trapping wells and barriers, and at different temperatures. The numericalcomputations determining the effective action are much faster than the direct simulation of thestochastic dynamics to which we compare our fRG results. We benchmark our numerical solutionsto the flow equations by comparing the first two equilibrium cumulants from the fRG against theBoltzmann distribution. We obtain excellent agreement between the two methods demonstratingthat numerical solutions for the effective potential can be accurately obtained in all the highlyunharmonic cases we examined. We then assess the utility of the effective potential to describethe equilibrium 2-point correlation function (cid:104) x (0) x ( t ) (cid:105) and the relevant correlation time. We findthat when Wavefunction Renormalisation is also utilized, these are obtained to percent accuracy fortemperatures down to the typical height of the potentials’ barriers but accuracy is quickly lost forlower temperatures. We also show how the fRG can offer strong agreement with direct numericalsimulation of the nonequilibrium evolution of average position and variance. Also, the fRG solutionrepresents the whole ensemble average, further adding to its convenience over other techniques,such as direct numerical simulations or solving the Fokker-Planck diffusion equation, which requiremultiple solutions with different initial conditions to construct averages over an ensemble. I. INTRODUCTION
In part one of this two part series we utilised one par-ticular formulation of the Renormalisation Group [1],namely the functional Renormalisation Group (fRG)[2,3]. We applied the fRG to Brownian Motion [4] and de-rived the flow equations [5] under two approximations ofthe fRG: the Local Potential Approximation (LPA) andWavefunction Renormalisation (WFR). We showed howthe solutions to these equations at k = 0 can have phys-ical meaning, firstly to static quantities such as equilib-rium position and variance, but also to dynamical prop-erties. In particular we derived for the first time EffectiveEquations of Motion (EEOM) for the evolution of the av-erage position, variance and covariance with full validityboth in and out of equilibrium. This is a completely newapproach in comparison to previous applications of thefRG to non-equilibrium physics [6–17]. In this secondpart we will present solutions to these equations frompart one and verify their validity and predictive powerfor physical processes with the bonus that the fRG routefor calculation is many orders of magnitude quicker thandirect simulation of the random walk. ∗ [email protected] † [email protected] ‡ [email protected] In Sec. II we present numerical solutions to the flowequations for five types of anharmonic bare potentialswhich represent highly non-trivial systems: a polynomialasymmetric potential which does not exhibit any localmaxima, a symmetric quartic double well, an asymmet-ric, double Lennard-Jones with two local minima and arather flat region around the maximum, as well as two“rugged” potentials consisting of a simple harmonic x potential with the addition of gaussian bumps. We con-sider different temperatures, ranging from relatively highto comparable to the depth of the potential’s barriers,controlling the amplitude of fluctuations. Naturally, wefind that the end results of the flow equations differ, withhigher temperatures resulting in potentials that carry lessmemory of the bare potentials’ morphology. Using theobtained effective potentials V k → we evaluate the meanposition of the particle (cid:104) x (cid:105) and its variance (cid:10) x (cid:11) − (cid:104) x (cid:105) at equilibrium and find excellent agreement with the ex-act results from the equilibrium Boltzman distribution.This confirms that the numerical solution of the LPA flowequation is accurate for the the wide variety of potentialshapes we have considered, at least around the minimaof the effective potentials.In Sec. III we verify that the equilibrium limit of theequations from part one are correct. In particular weverify that the LPA accurately predicts equilibrium posi-tion and variance correctly, which confirms the accuracyof our solution to the flow equations. We then examinethe characteristic decay behaviour of the connected 2- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b point function (cid:104) x (0) x ( t ) (cid:105) or covariance at equilibrium byutilizing both the effective potential V k → and the WFRfunction Z k → , which is instead a test of the accuracy ofthe fRG formalism itself and in particular the LPA andWFR approximations. We evaluate how these fRG pre-dictions compare to results from numerical simulations ofthe random walk and find very good agreement betweenthem down to relatively low temperatures comparable tothe height of the barriers in the potential. Where possiblewe also compare with the characteristic decay time ob-tained by exactly solving the corresponding Schr¨odingerequation (derived from the Fokker-Planck equation) forthe lowest eigenvalue. We find that the the LPA + WFRfRG equations give a comparable match to the simulateddecay rate at the percent level.In Sec. IV we extend our results to non-equilibriumevolution. We present numerical solutions to the EEOMfor the one- and two-point functions derived in part one.Our solutions to the EEOM are compared to both directnumerical simulation of equation (1) and the result fromthe evolution of the Fokker-Planck equation (31) wherepossible. We show the ability of the fRG to capture thenon-trivial non-equilibrium evolution in these various po-tentials to reasonable accuracy outlining a new, compet-itive method to compute accelerated dynamics for thissystem of interest.We conclude in Sec. V by summarising our main re-sults and discuss possible future directions. In AppendixA we explicitly present the WFR solutions to the flowequation (5). II. SOLUTIONS TO THE FLOW EQUATIONS
The physical system we are interested in is that of theoverdamped Langevin equation:˙ x = − ∂ x V ( x ) + η ( t ) (1) (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = Υ δ ( t − t (cid:48) ) (2)In this section we numerically integrate the flow equa-tions for this system obtained in part one. These are forLPA: ∂ k V k ( χ ) = Υ4 · k + ∂ χ V k ( χ ) , (3)and for WFR: ∂ k V k ( χ ) = Υ4 · k + ∂ χχ V k ( χ ) (4) ∂ k ζ ,χ = Υ4 · P ζ ,χ · D (5)We obtain the resulting effective potential and wave func-tion renormalisation for five types of potentials: i ) A simple polynomial: V ( x ) = x + x gx x g > ii ) The doublewell: V ( x ) = ax + bx (7)with a < b > iii ) A doublewell made by twoLennard-Jones (LJ) potentials back to back: V ( x ) = 4 (cid:15) (cid:18) σ ( x + 3) − σ ( x + 3) (cid:19) + 4 (cid:15) (cid:18) σ ( x − − σ ( x − (cid:19) (8)where σ will be taken to be 1 from here on in and (cid:15) & (cid:15) represents the depth of each well. E.g. if (cid:15) = (cid:15) = 1 bothwells are 1 unit deep (in 2 D /ε units) and the potentialis symmetric. Clearly here the domain of interest is x ∈ ( − ,
3) as the potential diverges at x = ±
3. We willalso consider the scenario of a simple x with additionalgaussian bumps (or dips): V ( x ) = x + n (cid:88) i =1 α i exp (cid:20) − ( x − β i ) µ (cid:21) (9)where there are n bumps or dips with the prefactor α i being positive or negative respectively. β i marks the lo-cation of each bump and µ the width of each bump whichfor simplicity we assume is the same for all. For our pur-poses we will focus on two variants of this setup: iv ) An x plus two bumps placed symmetrically away from theorigin and v ) an x plus 3 bumps and 3 dips in an asym-metrical setup. This potential represents a rudimentarytoy model for motion over a “potential energy landscape”with a series of local energy minima. The last two casesclearly demonstrate the effect of local extrema on the fi-nal shape of the effective potential since the underlying x potential does not alter its shape under the RG flow.The minima of the LPA effective potentials determinethe equilibrium position of the particle in each case andtheir curvature at the minimum determines the vari-ance. We find excellent agreement with the values ob-tained by the Boltzmann equilibrium distribution, al-lowing us to establish the accuracy of our numericalsolution for the LPA flow equation. In the LPA ap-proximation, the curvatures of the effective potentialsaround the minimum also determine the decay rates ofthe autocorrelation functions in equilibrium. We com-pare these with high accuracy numerical simulations ofthe stochastic process done using the open source soft-ware Large-scale Atomic/Molecular Massively ParallelSimulator (LAMMPS). We find that the LPA predictionalone leads to poorer agreement with simulations thanthe LPA + WFR prediction. In particular we find that,when wavefunction renormalisation is included, the au-tocorrelation decay rates for the potentials we study arequantitatively predicted by the fRG at the percent levelabove a certain temperature that we quantify.Unless otherwise stated our chosen parameters will be:Doublewell: a = − , b = 1 / g = 2 (11)Unequal L-J: (cid:15) = 1 , (cid:15) = 10 , σ = 1 (12) x + 2 bumps: α = α = 1 . , µ = 0 . , β = − β = 1 (13) x + 6 bumps/dips: α = α = α = − . α = α = α = 1 . , µ = 0 . β = − β = 0 . , β = − β = 1 . , β = − β = 2 . ε where we have chosen the ref-erence temperature 2 D = 1 /ε such that x = ˆ x . Thisprocedure essentially represents all physical quantities interms of the fundamental timescale ε . Concretely thismeans the doublewell potential is (restoring theˆfor clar-ity): ˆ V (ˆ x ) = − ε ˆ x + 14 ε ˆ x = ε (cid:20) − ˆ x + 14 ˆ x (cid:21) (15)As ε is by necessity a small number we will plot thesepotentials in units of ε later.
1. Polynomial Truncation
Before solving the full PDE (3) it is instructive to con-sider an approximation, focusing on the double well po-tential (7) for illustration. We consider a truncated poly-nomial ansatz for the effective potential V k ( χ ) of the form V k ( χ ) = E ( k ) + N (cid:88) i =1 α i ( k ) χ i (16)with initial conditions defined such that it matches theoriginal doublewell potential (7) at the cutoff: E ( k = Λ) = 0 (17) α ( k = Λ) = a (18) α ( k = Λ) = b (19)and all coefficients of higher powers vanishing. We canthen expand the r.h.s of (3) in powers of χ , truncate theseries at 2 N and therefore write a set of N + 1 coupledODEs in terms of the couplings that can then be solvednumerically. Below we write the set of ODEs for the O (4)truncation as it only concerns coupling coefficients up toorder x : dE ( k ) dk = Υ4 · (cid:18) k + 2 · α ( k ) − k (cid:19) (20) dα ( k ) dk = − · α ( k )( k + 2 · α ( k )) (21) dα ( k ) dk = 36Υ · α ( k )( k + 2 · α ( k )) (22) FIG. 1:
The convergence of the truncated system of ODEsto the full LPA PDE value for
Var ( x ) at equilibrium forΥ = 10 (red) and Υ = 1 (blue). Var ( x ) as calculated fromthe Boltzmann distribution is also included for reference These equations show how the coefficients in the polyno-mial ansatz for the potential evolve when fluctuations oflower and lower frequencies are averaged over. Keepingmore terms in the polynomial truncation is straightfor-ward, leading to a hierarchy of flow equations for thedifferent coefficients that can be easily obtained via acomputer algebra software. Solving such polynomial flowequations is numerically much easier than solving the fullPDE (3) and the solution to the full PDE should be ap-proached as N → ∞ . However, this method is only wellsuited to initial potentials of polynomial form of smalldegree (e.g. the doublewell − x + x / Var (x) = Υ / α ( k = 0), showing natu-rally how the different coupling constants relate to phys-ical quantities – the variance is inversely proportional to α ( k = 0).The results of this truncation for the doublewell poten-tial are displayed in Fig. 1. Here we can see that the low-est order truncations match poorly with the correct valueas given by the Boltzmann distribution. This discrepancybeing particularly noticeable for Υ = 1 with predictionsof negative variance which is unphysical. However thevalue calculated by solving the full LPA PDE (3) is ap-proached by including more terms with the O (20) trun-cation matching the full PDE at both temperatures. Theavailable thermal kinetic energy is E th = Υ / > < FIG. 2:
The flow of the polynomial Langevin potential V inthe LPA for Υ = 10 (High temperature/strong fluctuations -top) and Υ = 1 (Low temperature/Weak fluctuations -bottom). The blue curve indicates the bare potential whichis progressively changed, through green and yellow, into thered effective potential, as fluctuations are integrated out. to solving the flow, with the added bonus that it can besolved much quicker than the full LPA PDE. However, ifthe initial potential is not well approximated by a polyno-mial such as the Unequal-Lennard Jones, or our bumpypotentials, then one is forced to solve the full LPA PDE.Furthermore, computing the autocorrelation decay raterequires one to go beyond the LPA and include WFR,doubling the complexity of any polynomial truncation.We therefore now turn to the full PDEs, the numeri-cal solution to which is both feasible and accurate as wedemonstrate.
2. Full PDEs
We solve the LPA flow equation (3) on a grid in the χ direction, using Matlab’s built in ode45 or ode15s func-tion to evolve in the k direction, depending on the poten-tial. For most potentials ode45 – which is based on anadaptive step size Runge-Kutta method – was sufficient. FIG. 3:
The flow of the doublewell Langevin potential V inthe LPA for Υ = 10 (top) and Υ = 1 (bottom). The bluecurve indicates the bare potential which is progressivelychanged, through green and yellow, into the red effectivepotential, as fluctuations are integrated out. A similar approach was used for including (5). The nu-merical derivatives in the χ direction were based on afinite difference scheme using the Fornberg method witha stencil size of 5 for the potentials under study. Whileincreasing the grid size improves the accuracy of the nu-merical derivative it also increases the number of coupledODEs to be solved, making the integration much morecomputationally expensive. A balance must be drawndepending on the potential in question. We considered1001 points with x ∈ ( − ,
3) for the ULJ and x ∈ ( − , x = ˆ x (2 D = 1 /ε ) and the vertical axisis expressed in units of ε . We have also expressed k inunits of ε . This is done such that the plotted flow ofthe dimensionless potential ˆ V ˆ k (ˆ x ) looks the same as ofthe dimensionful potential V k ( x ). Therefore even thoughthis section will use V ( x ) and k to refer to the dimen-sionless parameters in units of ε and ε respectively theycould just as easily be the dimensionful versions whichwe explicitly obtain by setting ε → FIG. 4:
The flow of the unequal L-J Langevin potential Vin the LPA for Υ = 10 (top) and Υ = 1 (bottom). Again,the bare potential is denoted by the blue curve and the k = 0 effective potential by the red one. tive potential for high and low temperature, Υ = 10 andΥ = 1 respectively, is shown in Fig. 2 involving a poly-nomial potential. The flow in the range k ∈ (0 . , k → k to its effective incarnation for highΥ = 10 and low Υ = 1 temperature. Again, for thehigh temperature in the range k ∈ (0 . , k = 0 .
001 the energy barrier has gotten
FIG. 5:
The flow of the x potential with two additionalbumps for Υ = 10 (top) and Υ = 1 (bottom). As before, thebare potential is denoted by the blue curve and the k = 0effective potential by the red one. significantly smaller meaning that we have started to in-tegrate over fluctuations that drive the particle over thebarrier. Naturally, when k = 0 is reached the potentialis fully convex (as it must be by definition of Γ) withno barriers to overcome. Similar behaviour is obtainedwhere again we consider the lower temperature, Υ = 1.As one might expect it takes ‘longer’ in k evolution forthe barrier to disappear as fluctuations at each k scalehave less energy than their equivalent for the Υ = 10case. Of note is that not only is the evolution differentbut the final shape of V k =0 ( x ) is different for the two dif-ferent temperature regimes. For Υ = 1 it is clear thatthe potential is much flatter around the origin than forΥ = 10. This is suggestive of longer time scales requiredat lower temperatures to overcome the energy barrier andreach equilibrium. It also indicates longer times for theconnected 2-point function to decay, as we discuss below.Also of note is that for both cases the global minimumshifts from its degenerate values at ±√ x = 0. Thismakes physical sense as one expects that the particle willspend most of its time at the bottom of each well sothat its average position will be in the middle i.e. at the FIG. 6:
The flow of the x potential with 3 additionalgaussian bumps and 3 dips in the LPA for Υ = 10 (top) andΥ = 1 (bottom). As before, the bare potential is denoted bythe blue curve and the k = 0 effective potential by the redone. origin. This is suggestive of the fact that the minimumof the fully flowed potential V k =0 ( x ) should correspondto the equilibrium position of the particle. We showedthat this is indeed the case in part one and will verifythis numerically.As our third example we turn to a non-symmetric non-polynomial potential. Fig. 4 displays the evolution with k for an Unequal double Lennard-Jones (ULJ) poten-tial under the LPA for high (Υ = 10) and low (Υ = 1)temperature. Similarly to the double well case the en-ergy barriers get smaller and eventually disappear as k is lowered and V k =0 is fully convex. As one might ex-pect however, V k =0 is not symmetric. Furthermore, theminimum of V k =0 does not match the global minimumof the bare potential. At high temperature, Υ = 10, theeffective global minimum is located at x > k = 0 is very closeto the bare potential’s global minimum, suggesting thatthe particle is nearly always found here at equilibrium.We see that the form of the effective potential clearlyreflects the physical fact that, as the temperature is low-ered, the particle is more likely to be found in the globalminimum as it has less energy to escape and explore itssurroundings.We finally turn to our last two example potentials,consisting of a simple x potential with the addition oftwo Gaussian bumps placed symmetrically at x = ± x potential the fRG flow equation (3) yieldsno change beyond an unphysical shift by an overall ad-ditive constant. Again, the difference between the lowand high temperature cases is evident in the asymmet-ric case with the high temperature flow eradicating thepotential’s substructure, while the low temperature flowends up with a preferred equilibrium position, indicatingthe particle being more likely to be found near the globalminimum. III. EQUILIBRIUM RESULTS
Using the equations of static equilibrium quantities(23) & (27) derived in part one we can verify the validityof our numerical routines employed in the previous sec-tion by comparing to the predictions from the Boltzmanndistribution. After we have verified this we can compareour prediction for the evolution of the covariance in equi-librium given by (28) to direct numerical solutions of theLangevin equation (1).
A. Equilibrium position
As we discussed in part one the minimum of the ef-fective potential at k = 0 corresponds to the averageposition of the particle in equilibrium: ∂ χ V k =0 ( χ eq ) = 0 (23)Here we verify that this is indeed the case by comparingthe position of these minima to that computed directlyfrom the Boltzmann distribution. The agreement pro-vides a check of the accuracy of the numerical solutionto (3). FIG. 7:
The value of χ eq for different values of the thermalenergy Υ in the polynomial potential as calculated via theLPA. The original bare polynomial Langevin potential isplotted (not to scale) in blue for contextPotentials Υ Boltz LPAPolynomial 10 -0.9618 -0.962 -1.3227 -1.331 -1.5170 -1.52Unequal L-J 10 0.4854 0.4852 1.8522 1.851 1.8684 1.87 x plus 6 bumps/dips 5 0.0531 0.0552 0.1597 0.16 TABLE I: χ eq as calculated from the Boltzmann distributionand the LPA effective potential. Let us consider the (normalised) equilibrium Boltz-mann distribution defined in the standard way: P ( x ) = N exp (cid:18) − V ( x )Υ (cid:19) (24)where N is chosen so that (cid:82) ∞−∞ P ( x ) = 1. We can thencompute χ eq in the standard way from the equilibriumprobability distribution function: (cid:90) ∞−∞ x · P ( x ) = χ eq (25)Looking at Table. I we can see that the LPA matches theBoltzmann distribution extremely well for a wide rangeof different potentials across the range of temperatureswe examined. In Figs. 7 & 8 we have plotted the LPAprediction for the average position as the thermal energyΥ of the system is lowered and the equilibrium positionshifts closer to the original potential’s minimum. This isparticularly stark in Fig. 8 as it is clear at high tempera-ture the equilibrium position is in the middle of the twowells suggesting a roughly symmetric Boltzmann distri-bution. However as temperature is lowered the χ eq moves FIG. 8:
The value of χ eq for different values of the thermalenergy Υ in the Unequal Lennard-Jones potential ascalculated via the LPA. The original bare unequalLennard-Jones type Langevin potential is plotted (not toscale) in blue for context into the deeper well indicating that particles at equilib-rium at low temperatures would nearly always be in thisregion as one would expect. Table I verifies that the nu-merical solution to the LPA flow equation (3) is accuratein capturing this crucial physical aspect of the system atequilibrium. B. Variance
Potentials Υ Boltz LPAPolynomial 10 1.5690 1.56952 0.5931 0.58941 0.2938 0.2922Doublewell 10 2.2198 2.21992 1.6655 1.66551 1.7043 1.7042Unequal L-J 10 1.6858 1.68602 0.01426 0.019621 1 . · − . · − x plus 2 bumps 10 2.5317 2.57592 0.4145 0.41451 0.1554 0.1554 x plus 6 bumps/dips 5 1.2550 1.25513 0.7824 0.78132 0.5497 0.5499 TABLE II: Var (x) as calculated from the Boltzmann dis-tribution and the LPA effective potential.
The fRG also allows for the computation of the vari-ance of the equilibrium distribution, defined by: (cid:90) ∞−∞ ( x − χ eq ) · P ( x ) = Var ( x ) (26)with the fRG predicting it to be Var eq ( x ) = Υ2 V ,χχ | (27)Clearly, the variance is related to how flat the k = 0 po-tential is near the minimum, controlled by V ,χχ at theequilibrium point. Unsurprisingly, the bigger the curva-ture of the effective potential, the smaller the variancefor a fixed temperature. As the temperature is low-ered the equilibrium distribution is confined to a smallerand smaller region of the potential energy surface witha smaller variance. If variance changed linearly withtemperature we can see from (27) that V ,χχ | would notchange as temperature was varied. However this vari-ance does not generically scale linearly with temperaturewhich is why the k = 0 curves in Figs. 2, 3 & 6 are gener-ically flatter about the equilibrium point for Υ = 1 thanfor Υ = 10 in order to accommodate the fact that theequilibrium variance decreases by less than a factor of 10for Υ = 10 → V ,χχ near the equilibrium point must decrease as temperature is lowered. However in Figs. 4& 5 the Υ = 10 → → V ,χχ near the equilibrium point must increase as temperature is lowered resulting in a steeper curve at k = 0. The takeaway point is that lowering temperaturein a particular range can generically make the effectivepotential flatter or steeper around the minimum depend-ing on the scaling of variance with temperature in thatregime.Either way, once the fRG flow equations have beensolved, calculating the curvature of the effective potentialat the minimum is very straightforward. Our results aresummarised in Table. II and it is clear that the LPA offersvery good agreement for the variance of the equilibriumdistribution for all the potentials examined. This is asit should, see Appendix B or part one, and, conversely,offers a check that the numerical solution to the LPAequation is accurate. In the following subsection we willsee what else the effective potential can tell us aboutthe system that is not immediately available from theBoltzmann distribution. C. Covariance
In addition to the static variance at equilibrium, thecurvature of the effective potential around the minimumalso determines the time dependence of correlations inequilibrium, quantified by the time dependent covarianceor connected 2-point function
Cov eq ( x ( t ) x ( t )) = Υ2 V ,χχ | e − λ | t − t | (28) FIG. 9:
The decay of the (normalised) covariance (cid:104) x (0) x ( t ) (cid:105) C at equilibrium in a polynomial potential forΥ = 10 (top) and Υ = 1 (bottom) where λ is given in part one and for the LPA correspondsto V ,χχ | . Furthermore, now the solution to the WFR flowequation (5) for ζ ,χ also contributes, providing a correc-tion to the decay rate λ . The full numerical solutionsto (5) for the potentials considered here are described indetail in Appendix (A).In Table. III we collect the values of λ obtained us-ing the fRG under LPA & WFR for different Υ val-ues, higher or comparable to the typical depth or bar-rier heights of the different potentials, and compare thisdirectly to high accuracy numerical simulations of theLangevin equation (1) using LAMMPS. Where possiblewe also computed the first non-zero eigenvalue E bydiagonalising the Hamiltonian from the Schr¨odinger (orrescaled Fokker-Planck) equation in (31). We can clearlysee from Table. III that for our potentials the value ob-tained via the LPA tends to deviate by ∼ −
15% fromthe simulation value. Inclusion of the WFR factor ζ ,χ re-duces the deviation substantially error to ∼ −
5% fromthe value obtained in the simulations. We plot the decayof the covariance at equilibrium for our five potentials ofinterest in Figs. 9, 10, 11, 12 & 13 as calculated by fRGtechniques compared to direct numerical simulations ofthe langevin equation. For a polynomial potential, as
FIG. 10:
The decay of the (normalised) covariance (cid:104) x (0) x ( t ) (cid:105) C at equilibrium in a doublewell potential forΥ = 10 (top) and Υ = 2 (bottom) shown in Fig. 9, we can see how the decay rate as calcu-lated via the fRG for both LPA and LPA + WFR closelymatches the simulations at both high and low tempera-ture. Fig. 10 shows the decay in the doublewell which atΥ = 10 (top plot) shows great agreement with the simula-tion and Schr¨odinger calculation of λ with fRG methods.At Υ = 2 (bottom plot) of Fig. 10 however we can seethat the LPA is poorly capturing the correct decay rateand the improvement gained by including WFR offers ismuch more dramatic. The calculation of E from theSchr¨odinger equation has proved a non-trivial numericalexercise for our unequal Lennard-Jones potential henceits omission from Table. III and Fig. 11. Here the fRGoffers a very real advantage over more conventional meth-ods to calculating this decay rate as we do not have todevelop special numerical routines for every potential ofinterest, we simply solve the same two flow equations (3)& (5). We can see in Fig. 11 how the LPA + WFR decayrate closely matches the simulated decay at high temper-ature with the advantage of being calculated much morequickly than the direct simulation. Even just the LPAdecay at low temperature as seen in Table. III puts us inthe correct ballpark for the decay rate.For our x -plus-bumps potentials the decay rate is FIG. 11:
The decay of the (normalised) covariance (cid:104) x (0) x ( t ) (cid:105) C at equilibrium in the ULJ potential for Υ = 10 FIG. 12:
The decay of the (normalised) covariance (cid:104) x (0) x ( t ) (cid:105) C at equilibrium in an x plus two gaussian bumpspotential for Υ = 10 (top) and Υ = 1 (bottom) shown for two and 6 bumps/dips in Figs. 12 & 13 re-spectively and as in the ULJ case the computation ofeigenvalues for these potentials is a non-trivial exercise.We can see in the top plot (Υ = 10) of Fig. 12 that theLPA and WFR both in good agreement with simulationsand in the bottom plot (Υ = 1) the two decays correctly0 FIG. 13:
The decay of the (normalised) covariance (cid:104) x (0) x ( t ) (cid:105) C at equilibrium in an x plus 6 gaussianbumps/dips potential for Υ = 3Potentials Υ LPA WFR Sim E / x + 2 bumps 10 0.9705 0.9590 0.9322 —2 1.2063 0.9071 0.9960 —1 1.6088 1.1149 1.4245 — x + 6 b/d 3 0.9599 0.6334 0.7232 —2 0.9096 0.3888 0.4724 — TABLE III:
Value of the autocorrelation decay rateobtained for various potentials at different temperatures bydifferent methods. The LPA & WFR columns display λ/ bound the simulated decay – we will discuss this morein a moment. In Fig. 13 we can see that for Υ = 3the LPA and WFR predictions appropriately bound thesimulated decay with the simulations asymptoting to theWFR decay at late times. This indicates that even forhighly non-trivial systems where the simulated decay isvastly different from the bare x potential – see Table.III and compare to the x prediction for λ/ E to accurately describe covariance for all systems of in-terest. As LPA matches the decay rate predicted by theBoltzmann distribution and WFR is closer to the decaypredicted by E it is apparent why having both is highlyuseful and why it is nice to be able to get both in thesame framework. IV. ACCELERATED DYNAMICS OUT OFEQUILIBRIUM
In order to solve the equations of motion for the onepoint function χ ( t ) and two point function G ( t, t (cid:48) ) wemust first solve the PDEs for the LPA & WFR to obtainthe dynamical effective potential ˜ V and the function U .We will use the solutions obtained in Section II in order tocompute these parameters and then solve the appropriateEffective Equation of Motion (EEOM). A. The dynamical effective potentials
In part I we introduced the notion of the dynamicaleffective potential ˜ V :˜ V χ ( χ ) ≡ V χ ( k = 0 , χ ) , for LPA V χ ( k = 0 , χ ) ζ χ ( k = 0 , χ ) , for WFR (29)As the fRG guarantees that the fully effective potentialV will be convex this implies that the dynamical effectivepotential ˜ V will also be either fully or extremely close tofully convex for LPA and WFR respectively thus greatlysimplifying dynamical calculations. In the previous sec-tion we emphasised how the fRG LPA effective potentialgives us the Boltzmann equilibrium quantities such asequilibrium position and variance. However, away fromthe minimum of the effective potential the fRG givesus information that the near equilibrium Boltzmann as-sumption does not. To be concrete the (Gaussian) Boltz-mann distribution assumes that the potential is of theform: ˜ V Boltz ( x ) = Υ4 · Var eq ( χ − χ eq ) (30)where χ eq and Var eq are the equilibrium position andvariance respectively. We show in Fig. 14 how this ap-proximation can break down dramatically as one movesaway from the equilibrium position suggesting that thefRG captures well the dynamics far away from equilib-rium. In principle one could attempt to include higherorder cumulants of the Boltzmann distribution such asskewness and kurtosis into the effective potential, how-ever as we showed in part one the relationship betweenthese cumulants and higher derivatives of the effective1 FIG. 14:
Comparison of ˜ V for ULJ potential at Υ = 10 ascalculated using fRG methods LPA and WFR compared tothe Boltzmann “near-equilibrium” approximation given byequation (30). All potentials have been vertically shifted sothat their minima (corresponding to the equilibriumposition) coincide. potential is highly non-trivial and is cumbersome to in-clude.In Fig. 15 we show the evolution of ˜ V as k is loweredto zero – or equivalently as all the fluctuations are inte-grated out – for the asymmetric doublewell. We can seefor both the LPA (top plot) and WFR (bottom plot) howthe barrier gets smaller as fluctuations are integrated outuntil it completely disappears. The equilibrium positionis represented by the global minimum of the red curve(k = 0) and we can infer the speed of the evolution tothis equilibrium by the slope of the curve to it. Simi-lar behaviour can be seen for all the other potentials weconsider in this paper. The fact that the fully flowedpotential (red curve) is guaranteed to be (near) convexby definition of the EA Γ ensures that the dynamics weperform in it will be trivial to solve. This is what wecover in the following subsection.Where possible we will compare our results with thoseobtained by direct simulation of the Langevin equation(1) and by solving the Fokker-Plank (F-P) equation:Υ2 ∂ ˜ P ( x, t ) ∂t = (cid:18) Υ2 (cid:19) ∂ xx ˜ P ( x, t ) + U ˜ P ( x, t ) (31) B. Accelerated trajectories
From part I we know that the EEOM for average po-sition is given by: ˙ χ = − ˜ V ,χ ( χ ) (32)Having solved the appropriate flow equations to obtainthe dynamical effective potentials we can now perform FIG. 15:
The flow of the dynamical effective potential ˜ V for an initially assymetric doublewell potential defined as V = V DW + x/ dynamics in this effective potential. Given the dynamicaleffective potential ˜ V it only takes a couple of seconds toobtain the full trajectory of χ from some initial position x i = χ i to the equilibrium position. For the polynomialpotential we initialised the particle far away from theequilibrium position at x = 4. In Fig. 16 we show howthe average position of the particle changes with timeusing direct simulation of the Langevin equation (1)over 5000 runs, by numerically solving the F-P equation(31) and as calculated by the evolution in the dynamicaleffective potentials ˜ V given using the LPA and WFRmethods at Υ = 10. All four trajectories agree to avery high precision. This is perhaps not surprising asthe polynomial potential we consider is rather simple.What is more significant however is how well the fRGworks for the symmetric doublewell. In Fig. 17 we plotthe four trajectories where the particle for each starts atthe bottom of the right hand well ( x = √ FIG. 16:
The trajectory of the average position χ in apolynomial potential ˜ V by direct simulation & solving theEEOM (32) using LPA and WFR for Υ = 10 FIG. 17:
The trajectory of the average position χ in adoublewell potential ˜ V by direct simulation & solving theEEOM (32) using LPA and WFR for Υ = 10 simple LPA describes pretty well the evolution of χ ( t )towards the equilibrium point at χ eq = 0. When WFRis also included it matches the simulated trajectory veryclosely although not quite as closely as solving the F-Pequation (31). This is a non-trivial system and it isremarkable how well the fRG does to capture the correctdynamics.In Fig. 18 we plot the evolution of χ ( t ) for the ULJpotential where the particle begins in the smaller wellat x = − .
878 and moves towards its equilibrium posi-tion. We see as before that the LPA and WFR trajec-tories closely match the simulated trajectory in this casebounding it above and below. For this system it was im-possible to get convergent numerics for the evolution ofthe F-P equation (31) showing that the fRG can deriveimportant quantities even in highly non-trivial systems
FIG. 18:
The trajectory of the average position χ in a ULJpotential ˜ V by direct simulation & solving the EEOM (32)using LPA and WFR for Υ = 10 FIG. 19:
The trajectory of the average position χ for x potential plus two gaussian bumps ˜ V by direct simulation &solving the EEOM (32) using LPA and WFR for Υ = 2.gaussian bumps ˜ V by direct simulation, & solving theEEOM (33) for Υ = 2. The average position χ predicted fora simple x potential is displayed to highlight the non-trivialbehaviour the fRG is capturing where competing methods struggle. Similarly in Fig. 19the particle is initialised to the left of one of the gaus-sian bumps at x = − .
5. While the LPA offers little/noimprovement over the evolution in the bare x poten-tial, including WFR offers excellent agreement with di-rect simulations and the F-P solution. This ability of thefRG to capture the non-trivial evolution of average po-sition is also shown in Fig. 20 for the x potential plus6 bumps/dips which is a much more complex potentiallandscape. Here the LPA trajectory offers improvementover the evolution in the bare x potential by convergingto the correct equilibrium position and including WFRmore closely matches the true simulated trajectory. It3 FIG. 20:
The trajectory of the average position χ for x potential plus 6 gaussian bumps/dips ˜ V by direct simulation& solving the EEOM (32) using LPA and WFR for Υ = 3 is noteworthy that the fRG is able to reasonably capturethese difficult dynamics well – with significant time gains– in systems where the F-P solution is difficult to obtain.It is important to note the time advantage offered bythe fRG compared to direct numerical simulation or bysolving the F-P equation (31). For example if we con-sider the symmetric doublewell at Υ = 10 it takes ∼ ∼ ∼ V is obtained it is trivial to solve theEEOM (32) in a couple of seconds for any initial positionwhereas for both direct numerical simulation of (1) andsolving the F-P equation (31) one has to start again fromscratch. C. Evolution of Var(x)
For our accelerated trajectories we initialised the parti-cles at the exact same point every time. This means thatat t = 0 the probability distribution of the particles hadzero variance
Var (x) = 0. Using this as our initial con-dition we solved numerically the EEOM for the variancederived in part one:
Var ( x ) = Υ2 λP ( t ) ˜ Y ( t ) ˜ Y ( t )+ P (0) P ( t ) (cid:20) G − Υ2 λP (0) (cid:21) ˜ Y ( t ) (33)In Fig. 21 we show how the variance evolves with timefor the polynomial potential for Υ = 5. We can see thatthe LPA closely matches the numerical and F-P evolutionvery closely for the first 0.5 time units before departing FIG. 21:
The evolution of the variance
Var (x) in apolynomial potential by direct simulation, solving theFokker-Plank equation & solving the EEOM (33) for Υ =10. The equilibrium variance is also shown as calculatedfrom the Boltzmann distribution
FIG. 22:
The evolution of the variance
Var (x) in adoublewell potential by direct simulation, solving theFokker-Plank equation & solving the EEOM (33) for Υ =10. The equilibrium variance is also shown as calculatedfrom the Boltzmann distribution slightly although it still tends towards the correct equi-librium distribution.In Fig. 22 we show how the variance evolves with timefor a symmetric doublewell potential for Υ = 10. We cansee that the LPA gives us excellent agreement with thesimulated and F-P evolution. Considering how quicklythe LPA solution is computed compared to both the F-P and directly simulated evolution, this highlights thebenefit of fRG techniques for accelerated dynamics.Finally in Figs. 23 & 24 we show how the varianceevolves with time for an unequal Lennard-Jones and an x plus 6 gaussian bumps type potential respectively. Aswith the one-point function, the F-P was unable to re-4 FIG. 23:
The evolution of the variance
Var (x) in an ULJpotential by direct simulation & solving the EEOM (33) forΥ = 10 . The equilibrium variance is also shown ascalculated from the Boltzmann distribution
FIG. 24:
The evolution of the variance
Var (x) for x potential plus 6 gaussian bumps/dips potential by directsimulation, by solving the bare x case & solving the EEOM(33) for Υ = 3 . The equilibrium variance is also shown ascalculated from the Boltzmann distribution solve the statistics whereas the LPA matched well thesimulated trajectory, in both cases even capturing thevariance overshooting its equilibrium value. Demonstrat-ing this is particularly significant for Fig. 24 as the bare x evolution does not capture this behaviour and overalldescribes the evolution poorly, converging to the wrongequilibrium variance. This shows that the fRG can alsocapture non-standard evolutions of non-equilibrium sys-tems where competing methods are either ineffective,much slower, or both. V. SUMMARY
We presented the results of various numerical solu-tions to the fRG flow equations and, from these solutions,solved the Effective Equations of Motion (EEOM) bothin and out of equilibrium.We have shown that a polynomial truncation of theLPA approaches the LPA result after sufficiently manyterms. However, for initial potentials which are not wellapproximated by a polynomial, solving the full PDE (3)is viable and accurate. We showed that the full PDE canbe solved numerically using standard techniques on a va-riety of potentials exhibiting barrier structures and trap-ping wells. We also demonstrated the accuracy of the so-lutions, at least around the effective potential minimum,via comparisons to results from the Boltzmann distribu-tion. The minimum of the effective potential (arising as asolution to the LPA flow equation) determines the parti-cle’s average equilibrium position χ eq = lim t →∞ (cid:104) x ( t ) (cid:105) , whilethe effective potential’s curvature at the minimum deter-mines the equilibrium variance lim t →∞ (cid:104)(cid:10) x ( t ) (cid:11) − (cid:104) x ( t ) (cid:105) (cid:105) .We found that the computed effective potentials to giveexcellent results for these quantities, implying that theflow equations can be accurately solved. Beyond thesetime independent quantities, the curvature of the ef-fective potential also provides an approximation to thedecay rate of the equilibrium autocorrelation function (cid:104) x (0) x ( t ) (cid:105) C but with an error that reaches 10 −
15% forthe potentials we examined and for temperatures rela-tively high compared to the heights of the potential bar-riers. The LPA error increases substantially as the tem-perature is decreased. Inclusion of WFR improves thecomputation of the autocorrelation function in the cor-responding cases significantly, allowing for the determi-nation of the decay rate with an accuracy of ∼ − (cid:39) height of barriers.The above observations suggest that the accuracy withwhich the autocorrelation function is represented by thefirst and second orders in the gradient expansion, LPA+ the WFR correction, generically decreases with de-creasing temperature. This implies that the gradient ex-pansion of the fRG for studying thermal fluctuations isreliable only in the range from moderate temperatures(thermal energies comparable to the barrier heights), upto the very high temperature regime (where the smalllocal features of the potential are irrelevant anyway). Incontrast, at temperatures Υ below the height of the bar-riers/depth of the wells present on the potential function,the LPA+WFR approximations to the flow equation ap-pear to break down in their ability to predict (cid:104) x (0) x ( t ) (cid:105) C .This helps to quantify the thermal window where the fRGresults are both valid and non-trivial.We showed how the out of equilibrium evolution of χ for highly non-trivial systems, such as a symmetricdoublewell or an asymmetric polynomial, can be well de-scribed by fRG techniques in a fraction of the time re-5quired by both direct numerical simulation and solvingthe F-P equation. By comparing the evolution of χ aspredicted by the fRG in potentials based on x plus gaus-sian bumps to the bare x solution we have proved thatthe fRG captures non-trivial local features of the poten-tial. In fact the fRG can still offer reasonable approxi-mations even in systems where the F-P numerics fail toconverge.We have also shown how the fRG can closely match theout of equilibrium evolution of the variance for both poly-nomial and doublewell potentials even just using the LPAapproximation. For an Unequal Lennard-Jones type po-tential the LPA variance has reasonable accuracy and stillcaptures highly non-trivial behaviour such as the vari-ance overshooting its equilibrium value before settling toit. This is in a system for which the F-P equation failsto be resolved.While the fRG techniques we have outlined do not of-fer perfect agreement with simulations they can computenon-equilibrium evolution in a tiny fraction of the timethat direct numerics takes with the LPA in particular of-fering results over 2 orders of magnitude quicker. Also,once the fRG computes the dynamical effective poten-tial ˜ V all the dynamics for any initial condition can betrivially computed in a couple of seconds. This meansthat it if one wishes to calculate, for example, the evo-lution of an ensemble of initial conditions the fRG offersa massive advantage over direct numerical simulation orsolving the F-P equation, which would have to start anew full calculation for each initial condition.The difficulties faced obtaining results at very low tem-perature may indicate that the derivative expansion isinsufficient for capturing dynamics in the low tempera-ture regime. However, such a conclusion cannot be drawnsolely from the results presented in this manuscript; in-deed, one should investigate whether the convergenceproperties of the thermal problem may be improved bya using a regulator function different from the Callan-Symanzik regulator, r = k , used here. The recent find-ings of [19, 20] suggest that an appropriately optimizedregulator, which also excludes the regime ω > k fromcontributing to the flow, can ensure good convergenceproperties and a sizeable radius of convergence. Com-parisons with [19, 20] are non-trivial because the super-symmetry of the Brownian motion problem makes thestructure of the flow equations different to that of a sim-ple scalar theory. This is an important question to beresolved however.Anther possibility is that as the temperature is low-ered, the particle ends up being trapped in one of the lo-cal minima and the relevant dynamics is more akin to anescape problem, where the Kramers escape rate formulaapplies, rather than motion on an effective potential. Theeffective action may then be fundamentally non-local andnot subject to a meaningful derivative expansion. Wesuspect this low temperature regime would bear strongsimilarities to quantum tunnelling computations, involv-ing appropriate instanton-like solutions and the appro- priate functional determinants from fluctuations aroundthem. In that case an fRG study analogous to that de-veloped in [21–23] would apply. Study of the fRG flowequation might then allow for analytic estimates of thetemporal timescales required for escape over the barriers.Examining this regime further would also be an interest-ing direction for future investigation.In terms of computational effort and speed, the solu-tion of both the LPA and WFR PDEs offer significant ad-vantages over direct numerical simulation of the Langevinequation, averaged over enough realisations to gain accu-rate statistics even in the more simple case of the equi-librium limit. It would be interesting to compare withstandard techniques involving the Fokker-Planck equa-tion at equilibrium, as there are well-established ways forsolving the latter, and the accuracy obtained for compli-cated, landscape-type potentials with many bumps anddips. Obtaining accurate spectra in such complicated,non-polynomial potentials – even at equilibrium – is nottrivial. fRG methods can be systematically extended tomultiple degrees of freedom and therefore may offer clearcomputational advantages in studying the stochastic dy-namics of more than one degree of freedom, especially infield theoretical problems. These are systems where thecorresponding functional Fokker-Planck equation is oftennumerically intractable.Future work could examine if higher order approxima-tions beyond the WFR offer any advantage.Advances in the above directions may lead to progressin theoretically tackling a broad range of physical phe-nomena with large separation between fundamentaltimescales of thermal fluctuations and long emergenttimescales of macroscopic change, addressing what is nowa major barrier for predictive simulations across scientificand engineering disciplines including materials science[24], drug design [25], protein folding [26], and cosmology[27]. ACKNOWLEDGEMENTS
AW would like to thank George Stagg for his numeri-cal insight on solving the flow equations (3) and (5). AWis funded by the EPSRC under Project 2120421. GRwould like to thank Nikos Tetradis for very useful dis-cussions at the early stages of this project, Julien Ser-reau for providing much insight on fRG computationsand Gabriel Moreau for sharing his PhD thesis, contain-ing many new results on the application of the fRG tothe Langevin equation. We would also like to thank thereferees of an earlier version of this paper for very usefulcriticisms, comments and suggestions which helped im-prove this work substantially, especially bringing to ourattention the recent works [19, 20] and for prompting usto clarify the possible relation to [21–23]6
FIG. 25:
The flow of ζ x for the polynomial potential forΥ = 10 (top) and Υ = 1 (bottom). Appendix A: Solving the WFR flow equation
Including WFR does not change the evolution or finalshape of V k ( x ) making it irrelevant for time independentequilibrium quantities. However, as it corrects the decayrate of the correlation function it adds one more function ζ ,x to be evolved in k even for equilibrium computations.Its evolution with k for four of the potentials is shown inFigs. 25, 26, 27 & 28. As in the main body of the textwe have plotted this in units where x = ˆ x (2 D = 1 /ε )meaning k is expressed in units of ε . However unlike inthe main body where this choice means we express thevertical axis for the potential in units of ε , ζ is merely aredefinition of length so the two axes have the same units.At the start of the flow (blue curves) ζ ,χ = 1 with a non-trivial χ dependence developing as k →
0, shown by thered curve. Similar to the potential, as k is lowered it takeslonger for changes to happen. ζ x ( x ) at k = 0 for Υ = 1is much flatter than for Υ = 10. The evolution of ζ x ( x )with k for the unequal L-J potential is shown in Fig. 27.The behaviour is similar to the doublewell case exceptnow it is not symmetric with a larger peak for x > FIG. 26:
The flow of ζ x for the double well potential forΥ = 10 (top) and Υ = 1 (bottom) FIG. 27:
The flow of ζ x in an unequal L-J potential underWFR for Υ = 10. unequal L-J potential. In both cases they form aroundthe local minima of the bare V potential, e.g. for thedoublewell this is at x = ±√ ζ x ( x ) is even more complicated for x with two bumpsas shown in Fig. 28. Interestingly highly complicated7 FIG. 28:
The flow of ζ x for an x potential with twogaussian bumps for Υ = 10 (top) and Υ = 1 (bottom) structure appears part way through the flow (around k =0 .
01 and k = 0 .
001 for Υ = 10 and 1 respectively) beforebeing smoothed out as k →
0. Some of this structurestill remains at Υ = 1 around the origin for k = 0 assimilarly observed in Figs. 25, 26 & 27. Qualitativelysimilar behaviour is also observed for the flow of ζ x ( x )for x plus 6 gaussian bumps/dips – see Fig. 29.It is perhaps not surprising that including the runningof ζ x seriously complicates the numerics of the problem,equation (5) is more complicated than (3), however theeffect is significant. For example we were unable to ob-tain stable numerics for the evolution of ζ x in the unequalL-J potential for the low temperature Υ = 1 hence its omission. The calculations for ζ x also take significantlylonger than for V k alone as much smaller timesteps are re-quired to be within acceptable numerical tolerances. Asan example the calculation of V k for the unequal L-J took ∼ ζ x on the same machine took several hours.With more specialist numerical integrators tailor madefor these equations it is conceivable that computationscould be done quicker and ζ x could be calculated forpotentials and temperatures currently inaccessible usingproprietary software. The advantage of this approachcompared to competing methods is that optimising solv- FIG. 29:
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