Oscillating states of periodically driven anharmonic Langevin systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Oscillating states of periodically driven anharmonic Langevin systems
Shakul Awasthi ∗ and Sreedhar B. Dutta † School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, India (Dated: February 4, 2021)We investigate the asymptotic distributions of periodically driven anharmonic Langevin systems.Utilizing the underlying SL symmetry of the Langevin dynamics, we develop a perturbative schemein which the effect of periodic driving can be treated nonperturbatively to any order of perturbationin anharmonicity. We spell out the conditions under which the asymptotic distributions exist andare periodic, and show that the distributions can be determined exactly in terms of the solutions ofthe associated Hill equations. We further find that the oscillating states of these driven systems arestable against anharmonic perturbations. I. INTRODUCTION
The study of the effects of periodic forces on a varietyof systems, including classical [1, 2], quantum [3–7] andstatistical systems [8–19], has been of continual interestfor various and diverse reasons.Periodically driven many-particle systems, under cer-tain conditions, can exist in a state exhibiting periodicthermodynamic properties. It may not be possible touncover the features of this oscillating state either fromthe knowledge of the equilibrium properties of the corre-sponding systems in the absence of driving or by studyingthe effects of weak periodic forces on the thermodynam-ics of such systems. Instead it may be required to explorethe behavior of such systems by necessarily treating thedriving nonperturbatively.A suitable framework to investigate the properties ofan oscillating state is presumably some sort of stochasticthermodynamics, wherein the periodic driving is appro-priately incorporated. The fluctuations of a macroscopicvariable of a thermodynamic system can be assumed tofollow a continuous Markov process even in the presenceof driving. It is expected that the stochastic process thatdescribes the fluctuations is continuous due to the macro-scopic nature of the variable. On the other hand, it is notevident whether or not the Markov property is a reason-able assumption. This is due to the fact that, as thefrequency of the driving increases, the contributions dueto the higher order time-derivative terms can become in-creasingly significant. Hence these higher order termsmay have to be accommodated in effecting the stochasticprocess. Essentially, we could include certain additionalvariables, apart from the ones that are relevant in theabsence of driving, and then assume that the extendedset of variables follows continuous Markov process andthat its asymptotic distribution describes the oscillatingstate.The study of an underdamped Brownian motion, sub-jected to periodic driving, can be of considerable inter-est for multiple reasons. First, it is a prototypical ex- ∗ [email protected] † [email protected] ample of the Langevin dynamics. Since any continuousMarkov process is represented by a Langevin equation,it may be possible to draw useful analogies between themacroscopic variables that follow such a process and theposition and velocity variables of the Brownian particle.Second, the velocity variable can be thought of as anadditional degree of freedom that is relevant when driv-ing is introduced to the overdamped Brownian motion.The extent of the relevance, for instance, can be esti-mated by comparing the marginal distribution of the un-derdamped motion, wherein the velocity is eliminated,with the distribution of the overdamped motion. In thespecial case of linearly driven Langevin equation, the cor-relation between the position and velocity degrees can be-come significant, depending on the driving [20]. Third,we can track almost analytically the underdamped Brow-nian motion even in the presence of driving, and hencethe study may prove useful in finding some generic fea-tures of the oscillating state.In an earlier work [20], we found that any linearlydriven Langevin equation can be solved exactly upon ex-ploiting the underlying SL symmetry. The exact solu-tion could reveal the presence of oscillating states undercertain conditions, could expose interesting properties ofsome observables and even could establish certain rela-tions among them. This motivates us to extend the en-quiry beyond linear driving and search for solutions, ifpossible exactly, in the presence of anharmonic pertur-bations. It is also natural to wonder whether we couldcapitalize on the symmetry even in the nonlinear case.We essentially ask the following central questions. Dodriven Langevin systems with anharmonic perturbationsreach an oscillating state asymptotically? Can we findthe probability distributions of the oscillating state ex-actly? What are the conditions under which these oscil-lating states exist and are stable?The layout of the current work is as follows. In the nextsection, we consider a generic class of driven nonlinearLangevin equations and formulate a perturbation schemewherein the underlying SL symmetry is not only mani-fest but also can be exploited to obtain the correspond-ing asymptotic distributions to any order. In Sec. III,we explore the conditions under which the driven sys-tems reach an oscillating state and can remain stable un-der perturbations. Finally, we summarize and concludebriefly with comments and remarks in Sec. IV. II. DRIVEN UNDERDAMPED BROWNIANPARTICLE: ASYMPTOTIC DISTRIBUTION
We will assume that the dynamics of a Brownian par-ticle, when subjected to periodic driving, is governedby the Langevin equation of an underdamped Brown-ian motion with time-dependent parameters that are T -periodic. If we view the Langevin dynamics as an effec-tive description, that is obtained upon eliminating thebath degrees of freedom, then it is reasonable to assumethat both viscous coefficient and the noise strength willhave to vary with the same period. Though there isno compelling reason for the external potential to varycommensurably, we nevertheless will restrict all time-dependent modulations to the same period. In this sec-tion, we shall obtain the asymptotic distribution of theperiodically driven underdamped Brownian particle, byperturbatively treating the nonlinear part of the externalpotential, while nonperturbatively accounting the peri-odic time dependence. A. Driven nonlinear Langevin dynamics
We shall consider the stochastic dynamics of the po-sition X t and velocity V t of a driven Brownian particle,governed by the following set of equations,˙ X t = V t , ˙ V t = − γV t + f ( X t , λ ) + η ( t ) , (1)where the viscous coefficient γ , the set of param-eters λ of the external force f and the Gaussiannoise η have T -periodic time dependence. Thenoise is assumed to have zero mean and nonzerovariance h η ( t ) η ( t ′ ) i η = 2 D ( t ) δ ( t − t ′ ), where the time-dependent diffusion coefficient D is T -periodic. The ex-ternal force f may also contain nonlinear terms of X t .The associated Fokker-Planck (FP) equation for theprobability distribution P ( x, v, t ) of the above nonlinearLangevin dynamics is given by ∂∂t P ( x, v, t ) = L ( x, v, g ( t )) P ( x, v, t ) , (2)where g denotes all the parameters, including γ and D ,and the FP operator L is defined as L ( x, v, g ) := − ∂∂x v − ∂∂v [ − γv + f ( x, λ )] + D ∂ ∂v . (3)It should be remarked that though it is common to as-sume the form of the external force to be f = f ( x, λ ),the analysis that we shall employ to obtain the asymp-totic distribution is oblivious to this restriction and in fact holds good for f = f ( x, v, λ ). Essentially, it is suffi-cient to assume that f is Taylor expandable in x, v vari-ables and thus can be written as f = − P n,m λ n,m x n v m ,where n, m are non-negative integers, and that the timedependence of the external force is implicitly providedby the linear parameter k and the nonlinear parame-ters λ n,m . We will tune off the parameter λ , to zero,since the linear term γv is already included, and denotethe linear coefficient λ , as k , for notational familiarity.The asymptotic distribution of the FP equation for pe-riodically driven Brownian particle can be expected to be T -periodic under certain conditions. In other words, thedriven stochastic system can exist in a state, that we referto as an oscillating state, wherein various relevant observ-ables exhibit periodic properties. Suppose we considerthe domain D eq in the parameter space γ, D, k and λ n,m which is defined as the set of all points for which theasymptotic distribution of the FP equation, when the pa-rameters are time independent, is an equilibrium distri-bution. Let us consider the domain D eq to be simply con-nected and the equilibrium state therein to be continu-ous. If we now drive the parameters of the FP equation inthis domain continuously, with a period much larger thanthe corresponding equilibrium relaxation timescales, thenthe asymptotic state essentially will be a periodic trajec-tory in the equilibrium state space. As we decrease theperiod, the trajectories of course will not be restricted tothe equilibrium state space but are likely to be continuousin some extended space, wherein the closed trajectoriespresumably can characterize the oscillating states of adriven system. While it is far from obvious what thisextended space is or how to characterize the oscillatingstates, we shall see that it may still be possible to spec-ify some of the conditions under which these states canexist.The asymptotic distribution is of course obtained bytaking the large time limit of the solution of the FP equa-tion [20]. We could choose the prescription for approach-ing the limit in periodically driven systems, for instance,by first decomposing time, t = N T + τ , in terms of aninteger N number of periods and a remainder τ suchthat 0 ≤ τ < T , and then by taking N → ∞ limit. Thusthe large time limit of the solution P ( x, v, t ) = U ( x, v ; t ) P ( x, v, , (4)of the FP equation (2), results in a T -periodic asymptoticdistribution P os ( x, v, τ ) = U ( x, v ; τ ) P ∞ ( x, v, , (5)provided the time-independent asymptotic distribu-tion P ∞ ( x, v,
0) exists. In the above equation, U denotesthe evolution operator, which is formally expressed as U ( x, v ; t ) := T (cid:8) e R t L ( x,v,g ( t )) dt (cid:9) , (6)where T indicates time ordering, and the time-independent distribution P ∞ ( x, v,
0) := lim N →∞ [ U ( x, v ; T )] N P ( x, v, , (7)where P ( x, v,
0) is the initial distribution. The existenceand uniqueness of P ∞ ( x, v,
0) is dictated by the spectrumof the monodromy operator U ( x, v ; T ). In case thereexists a unique distribution then it necessarily satisfiesthe eigenvalue equation U ( x, v ; T ) P ∞ ( x, v,
0) = P ∞ ( x, v, . (8)When the parameters are chosen from the domain D eq with their values kept fixed in time, then the above equa-tion is equivalent to L P ∞ = 0 and the asymptotic dis-tribution is an equilibrium distribution P eq ( x, v ). Hencewe could equivalently define the domain D eq as the setof all g for which the FP operator L ( x, v, g ) is negativesemi-definite, with a unique normalizable eigenfunctioncorresponding to the zero eigenvalue, and with a nonzerogap between zero and the real part of any other eigen-value. These properties of the FP operator are also re-quired for the equilibrium state to avoid any singularbehavior in D eq . Suppose we now drive the parame-ters g = g ( t ) piecewise continuously, though not nec-essarily respecting the negative semi-definite propertyof L ( x, v, g ( t )), but necessarily ensuring that the realpart of none of the eigenvalues of U ( x, v ; T ) exceedsunity. Then under such conditions the time-independentasymptotic distribution P ∞ ( x, v,
0) can exist, and hencethe system can reach an oscillating state described by thedistribution P os ( x, v, τ ). In some sense, the domain D os of the parameter space for which the oscillating statesexist can be larger than the domain D eq for which theequilibrium states exist. This may even suggest that thestability of the states of macroscopic systems can presum-ably be controlled and manipulated by periodic driving.In order to comprehend the role of driving in the os-cillating states, it is natural to ask the following ques-tions. What are the solutions of the FP equation fora generic driving g ( t )? Do these solutions have a largetime limit? Namely, does the system reach an oscillat-ing state independent of the initial distribution? Can wespecify the necessary conditions that guarantee an os-cillating state without having to solve explicitly the FPequation? In other words, can we find some criteria thattells us whether or not a given g ( t ) is in the domain D os ?In case of linear external force, using certain techniquesfrom representation theory, we could find almost exactlythe oscillating state and establish rigorously the neces-sary conditions for its existence [20]. We shall now show,even in cases where the external forces contain nonlinearterms, that similar techniques can be employed to an-swer these questions, provided we deal with nonlinearityperturbatively. B. Perturbative analysis to all orders
Following the standard perturbative analysis, we candecompose the FP operator, L = L + ǫ L p , (9) into a solvable part that includes only the linear forceterms, given by L = L ( x, v, g ) = − ∂∂x v + ∂∂v [ γv + kx ] + D ∂ ∂v , (10)where g denotes γ, D and k , and a perturbative part L p = L p ( x, v, λ ) := ∂∂v O p , (11)where O p = O p ( x, v, λ ) = − kx − f ( x, v, λ ) consists onlythe nonlinear terms of the force f . The dimensionlessparameter ǫ is introduced to conveniently track the or-der of nonlinearity. Expanding the probability distribu-tion P ( x, v, t ), in the FP equation (2), as the followingseries in the perturbative parameter ǫ , P ( x, v, t ) := ∞ X n =0 ǫ n P ( n ) ( x, v, t ) , (12)leads to a sequence of dynamical equations for P ( n ) .The zeroth-order equation is given by ∂∂t P (0) ( x, v, t ) = L ( x, v, g ( t )) P (0) ( x, v, t ) , (13)which is the FP equation of a driven Brownian particlein harmonic potential [20]. The other higher-order cor-rections P ( n ) are governed by the recursive equations, ∂∂t P ( n ) = L P ( n ) + L p P ( n − , (14)for any integer n ≥
1. The dependence on the coor-dinates x, v, t and on other parameters is not explicitlyexhibited here for notational simplicity.We will assume P ( x, v, t ) is normalized to any givenorder of ǫ . In other words, P (0) ( x, v, t ) is normal-ized, and R dxdvP ( n ) ( x, v, t ) = 0 for all n ≥ P ( x, v, t ) as P ( n ) ∞ ( x, v, t ), which are obtained by tak-ing the large time limit of the solutions of Eqs. (13)and (14). We could choose the initial conditions forthese equations such that the initial zeroth-order distri-bution P (0) ( x, v,
0) = P ( x, v, P ( n ) ∞ ( x, v,
0) = 0, for n ≥ P (0) ∞ ( x, v, t ) is in facta Gaussian distribution with zero mean and a T -periodiccovariance matrixΣ( t ) := (cid:20) h x i h xv i h vx i h v i (cid:21) ≡ (cid:20) e X , ( t ) e X , ( t ) e X , ( t ) e X , ( t ) (cid:21) , (15)whose matrix elements depend on the parameters g ( t ).The methods employed to find the conditions for the ex-istence of the asymptotic limit and to obtain the distri-bution P (0) ∞ ( x, v, t ) are detailed in the Ref. [20]. Sincewe shall extend these methods to obtain the higher-order asymptotic functions P ( n ) ∞ ( x, v, t ), for n ≥
1, wewill briefly spell them out, as and when required.The algorithm that we shall employ to evaluate theasymptotic n -th order correction is as follows. We firstsubstitute P ( n − in the equation (14) with its asymp-totic function P ( n − ∞ which we assume exists, and thenfind the solution of the modified equation. This solu-tion is valid only for times that are large compared tothe time required for the ( n − t = ( N + N ) T + τ and solve Eq. (14)for t = N T + τ with the new initial condition given attime N T , where N is chosen such that the substitu-tion P ( n − with P ( n − ∞ is justifiable, and then takethe limit N → ∞ . We can then show that the asymp-totic limit P ( n ) ∞ ( x, v, t ) of the solution to the modifiedEq. (14) exists when a certain condition holds for any ar-bitrary initial function P ( n ) ( x, v, N T ). We shall discusslater and analyse in detail this specific condition whichis in fact independent of n . Thus when this conditionholds and the zeroth-order distribution P (0) ∞ ( x, v, t ) ex-ists, we establish iteratively the existence of P ( n ) ∞ ( x, v, t )for any n > n -th or-der correction is similar to those for determining the first-order correction. We begin by rewriting the first-ordercorrection P (1) ( x, v, t ) = − (cid:16) A (1) − h A (1) i (cid:17) P (0) ∞ ( x, v, t ) , (16)where A (1) = A (1) ( x, v, t ) is yet to be determined func-tion and h A (1) i is the average of A (1) with respectto P (0) ∞ ( x, v, t ). If the perturbation O p ( x, v, λ ) is a poly-nomial function in x and v then A (1) ( x, v, t ) will also bea polynomial in these variables. Owing to the underly-ing SL symmetry of the unperturbed FP equation, it isbeneficial to choose the basis functions O rL := x L − r v r , (17)where L is a positive integer and denotes the degree ofhomogeneity and r , for a given L , runs over the integersfrom 0 to L . Let us consider the perturbation O p to bea nonlinear polynomial of degree L p , and hence can bewritten in the form O p ( x, v, λ ) = L p X L =2 L X r =0 λ Lr O rL , (18)where the parameters λ Lr , in case they depend on time,are T -periodic. We can represent the function A (1) inthis basis as A (1) ( x, v, t ) = L X L =1 L X r =0 a Lr ( t ) O rL , (19) where the coefficients a Lr ( t ) are time dependent and L is a finite positive integer such that L ≫ L p . We shallhenceforth refer the coefficients of O rL in an expansion aslevel- L coefficients.We now proceed to determine the coefficients a Lr ( t ).Substituting Eqs. (16), (18) and (19) in the recursiveequation (14), for n = 1, straightforwardly leads to apolynomial equation. Then equating the coefficients ofeach monomial in the polynomial equation results in anordinary differential equation for the variables a Lr . Thesedynamical equations can be written in the form ddt a Lr = H Lr + N Lr + R Lr + S Lr , (20)where the first term H Lr contains only the level- L coeffi-cients of A (1) and is given by H Lr = − ( L + 1 − r ) a Lr − + rγ p a Lr + ( r + 1) k p a Lr +1 , (21)and include the T -periodic parameters γ p := γ − D (Σ − ) ,k p := k − D (Σ − ) ; (22)the second term N Lr contains only a level-( L + 2) coeffi-cient of A (1) and is given by N Lr = ( D ( r + 1)( r + 2) a L +2 r +2 , for 1 ≤ L ≤ L − , , for L ≥ L − R Lr contains only a level-( L + 1) coefficientof the perturbation O p and is given by R Lr = ( − ( r + 1) λ L +1 r +1 , for 1 ≤ L ≤ L p − , , for L ≥ L p ; (24)while the fourth term contains only level-( L −
1) coeffi-cients of the perturbation O p and is given by S Lr = ( Σ − λ L − r + Σ − λ L − r − , for 3 ≤ L ≤ L p + 1 , , for either L ≥ L p + 2 or 1 ≤ L ≤ . (25)Furthermore, the constant term of the polynomial equa-tion leads to an additional equation ddt h A (1) i + 2 Da = 0 , (26)which though is not an independent one. Note that thedynamical equations (20) are such that the level- L coef-ficients a Lr are all linearly coupled amongst themselves.Further they contain the inhomogeneous terms involvingboth the variables that are level-( L + 2) coefficients a L +2 r +2 and the given interaction terms R Lr and S Lr . The inter-action terms though are absent for the dynamical equa-tions with L ≥ L p + 2. Hence it is evident that we needto solve for a L +2 r +2 in order to solve for a Lr .In order to solve Eqs. (20) which are inhomogeneous, itis useful to first discuss the symmetries and solutions ofthe corresponding homogeneous equations obtained fromEqs. (20) by dropping N Lr , R Lr and S Lr terms. The ho-mogeneous equations can be rewritten as ddt a L = h − J − L + γ p LI L + J L ) + k p J + L i T a L , (27)where a L is a ( L + 1)-component vector whose elementsare a Lr , namely a TL := (cid:0) a L , a L , · · · , a LL (cid:1) , the superscript T on a vector or a matrix denotes their transpose, thematrix I L is the identity matrix of dimension ( L + 1),and { J ± L , J L } are matrices of the same dimension withmatrix elements (cid:0) J + L (cid:1) r,s = rδ r,s +1 , (cid:0) J − L (cid:1) r,s = ( L − r ) δ r,s − , ( J L ) r,s = (2 r − L ) δ r,s , (28)and whose indices r, s run over all the integers from 0to L . We can easily verify that these matrices satisfy thecommutation relations of the sl algebra, namely (cid:2) J L , J ± L (cid:3) = ± J ± L , (cid:2) J + L , J − L (cid:3) = J L . (29)In fact these matrices form an irreducible representationof the generators of the group SL ( R ). This dynami-cal symmetry exhibited by the homogeneous equations isindeed induced by the underlying SL symmetry of theunperturbed FP operator.It may be remarked in passing that Eq. (27) can bemapped to a transposed version by a similarity trans-formation induced by an anti-diagonal matrix S withmatrix elements S r,s = δ r + s,L r !( L − r )! /L !. Under thistranspose map J + L and J − L reverse their role in the alge-bra as is evident from the relations S − J L S = − ( J L ) T and S − J ± L S = (cid:0) J ± L (cid:1) T .The explicit L -dependent term in Eq. (27) can be re-moved by writing the vector a L in terms of another vec-tor b L = a L exp( − L Γ p / p ( t ) = R to dt ′ γ p ( t ′ ),and thus obtain the equation ddt b L = h − J − L + γ p J L + k p J + L i T b L , (30)which has a form independent of any specific sl repre-sentation. This form makes it amenable to determineits solutions by exploiting the theorem which states thatany irreducible representation of sl is a symmetric powerof the standard representation [21]. In other words,we obtain the solutions of Eq. (30) by taking the sym-metrized tensor products of the solutions of its corre-sponding equation in the standard representation, whichis ddt b = h − J − + γ p J + k p J +1 i T b , (31)or, when written in terms of components of the vec-tor b T := ( b , b ), is ddt (cid:18) b b (cid:19) = (cid:18) − γ p k p − γ p (cid:19) (cid:18) b b (cid:19) . (32) Notice that the component b satisfies the Hill equation d dt b + ν p b = 0 , (33)where the T -periodic parameter ν p = k p −
12 ˙ γ p − γ p . (34)Hence the solutions of Eqs. (31), (30) and (27) can besolely expressed in terms of the two independent solutionsof the Hill equation, denoted u ( t ) and w ( t ), which can bechosen to be pseudo-periodic with Floquet exponents µ p and − µ p , respectively, namely u ( t + T ) = u ( t ) exp( µ p T )and w ( t + T ) = w ( t ) exp( − µ p T ). The psuedo-periodicsolutions of Eq. (31) are b (0)1 = (cid:18) γ p u − ˙ uu (cid:19) , b (1)1 = (cid:18) γ p w − ˙ ww (cid:19) , (35)while those of the homogeneous equation (27) are a ( r ) L = e L Γ p Sym (cid:20)(cid:16) b (0)1 (cid:17) ⊗ ( L − r ) ⊗ (cid:16) b (1)1 (cid:17) ⊗ r (cid:21) , (36)where r runs over the integers from 0 to L and Symdenotes the symmetrization of the tensor products of thevectors in the bracket. The solutions a ( r ) L are pseudo-periodic with corresponding Floquet exponents µ ( r ) L = 12 Lγ p + ( L − r ) µ p , (37)where γ p is the time average of γ p over a period T , andthey vanish in the large-time limit provided the modulusof the real part of the fundamental Floquet exponent | Re ( µ p ) | < − γ p . (38)Henceforth we will assume that the parameters k, γ and D are chosen such that the condition (38) holds. Wecan now deduce the large-time solutions a Lr ( t ) of Eq. (20)for each L sequentially in the reverse order startingfrom L = L down to L = 1. For L p + 2 ≤ L ≤ L , theset of equations (20) are homogenous and hence the cor-responding large-time solutions a Lr ( t ) vanish for any arbi-trary initial conditions. The equations (20) for L = L p +1and L = L p are though inhomogeneous have only given T -periodic inhomogeneous terms h Lr = R Lr + S Lr . Thesolutions of these equations can formally be written as a L ( t ) = K L ( t, a L (0) + Z t dsK L ( t, s ) h L ( s ) , (39)where for any L the vector h L is defined by the compo-nents h Lr , and the matrix K L ( t, s ) = Φ L ( t )Φ − L ( s ) (40)is defined by the fundamental matrix Φ L ( t ) of Eq. (27)constructed with a ( r ) L ( t ) as column vectors. Nowto determine the large-time limit of the solutions,we use the following two properties of the ma-trix K ( t, s ). First, the matrix can be decomposedas K L ( t, s ) = K L ( t, t ′ ) K L ( t ′ , s ) for any t ′ , which isa consequence of its definition. Second, it is in-variant under discrete time translation by a period,namely K L ( t + T, s + T ) = K L ( t, s ), which is due tothe pseudo-periodic nature of the fundamental ma-trix, Φ L ( t + T ) = Φ L ( t )Λ L , where Λ L is a diagonal ma-trix with elements exp( µ ( r ) L T ). Using these properties,for any time t = N T + τ the first term on the right handside of Eq. (39) can be written as K L ( τ,
0) [ K L ( T, N a L (0) , (41)while for any T -periodic function h L ( t ) the second termcan be written as K L ( τ, (cid:0) − K L ( T, N (cid:1) Z L ( T ; h L ) + Y L ( τ ; h L ) , (42)where the matrices Y L ( τ ; h L ) and Z L ( T ; h L ) are inde-pendent of N and are defined for any given T -periodicvector f L ( t ) as Y L ( τ ; f L ) = Z τ dsK L ( τ, s ) f L ( s ) ,Z L ( T ; f L ) = (1 − K L ( T, − Y L ( T ; f L ) . (43)Note that K L ( T,
0) = Φ L (0)Λ L Φ − L (0), and hence itseigenvalues are same as those of Λ L , namely exp( µ ( r ) L T ).Now when the condition (38) holds not only the ma-trix Z L ( T ; h L ) is ensured to be nonsingular but also thematrix K L ( T, N approaches zero in the limit N → ∞ .Consequently, a well-defined asymptotic limit of the so-lution that is independent of the initial conditions exists,and we obtain the asymptotic solution as a L ( τ ) = K L ( τ, Z L ( T ; h L ) + Y L ( τ ; h L ) , (44)for L = L p +1 and L = L p . It is straightforward to verifythat a L ( τ + T ) = a L ( τ ). This also implies that N Lr ( t )becomes T -periodic for L = L p − L = L p −
2. Itshould emphasized that the asymptotic solution (44) is T -periodic in spite of the psuedo-periodic nature of thehomogeneous part of the solution (39).By hierarchically repeating the arguments that lead toEq. (44), we further obtain the other T -periodic asymp-totic solutions a L ( τ ) = K L ( τ, Z L ( T ; h L + n L )+ Y L ( τ ; h L + n L ) , (45)for 1 ≤ L ≤ L p −
1, where the components of the vec-tor n L are defined to be the asymptotic values of N Lr .We can of course determine the next-order correc-tion P (2) ∞ ( x, v, t ) by exactly going through the same math-ematical manipulation as performed earlier to deter-mine P (1) ∞ ( x, v, t ) after replacing the nonlinear term O p with ( h A (1) i − A (1) ) O p . Thus proceeding iteratively, cor-rection term P ( n ) ∞ ( x, v, t ) to any order n can be obtained. To summarize this section, we have shown that theasymptotic probability distribution of the periodicallydriven nonlinear FP equation (2) to any order of per-turbation in anharmonicity exists and is T -periodic pro-vided the unperturbed oscillating state exists and thecondition (38) holds. Furthermore all the coefficients ofthis T -periodic asymptotic distribution to any order canbe determined exactly in terms of the solutions of theHill equation (33). III. STABILITY OF THE OSCILLATING STATE
In this section, we study the stability of the oscillatingstates and the effect of the perturbations. In particular,we will survey the domains of the unperturbed oscillat-ing states and then examine whether perturbations cancoexist within these domains.
A. Unperturbed oscillating states
The zeroth-order asymptotic distribution P (0) ∞ ( x, v, t )exists when the first moments vanish at large times andthe asymptotic covariance matrix is positive definite [20].The first moments of the distribution P (0) ( x, v, t ) are re-lated to the solutions Y of the following Hill equation d dt Y + νY = 0 , (46)where ν = k − ˙ γ/ − γ /
4. When the Floquet expo-nents ± µ of this Hill equation satisfy the condition | Re ( µ ) | < γ , (47)then the first moments vanish asymptotically. This con-dition also means that exp( − γt/ Y ( t ) should remainbounded at all times. Furthermore the condition (47) en-sures that the second moments also remain bounded andbecome T -periodic asymptotically independent of the ini-tial conditions of the distribution. We could deduce thisfact since the second moments also can be written interms of the Floquet solutions of Eq. (46). Essentially thedynamical equations of the second moments X , , X , and X , are ddt X , = 2 X , ,ddt X , = − kX , − γX , + X , ,ddt X , = − kX , − γX , + 2 D , (48)and the moments X , , X , and X , approach theirasymptotic values e X , , e X , and e X , , respectively, inthe large time limit. The homogeneous part of theseequations with parameters k and γ has similar structureas Eq. (27) for L = 2 with corresponding parameters k p and − γ p . Hence the manner in which Eq. (27), its solu-tion (36) and its exponents (37) are related to the Hillequation(33) is the exact manner in which the homoge-neous part of Eq. (48), its solution and its exponents arerelated to the Hill equation (46). In fact the Floquet ex-ponents associated with the covariance matrix can alsobe read from Eq. (37) and are − γ + (2 − r ) µ , where r = 0 , , k and γ implicitly, and hence itis far from obvious without explicit verification whethera given driving allows the system to be in an oscillat-ing state. We can rewrite the condition in a form thatis more amenable to computation using Floquet theoryof the Hill equation. The Floquet exponents essentiallycan be expressed in terms of the solutions of the Hillequation at time T [22, 23]. Let u ( t ) and u ( t ) be twoindependent solutions of the Hill equation (46) with theinitial conditions: u (0) = 1, u (0) = 0, ˙ u (0) = 0 and˙ u (0) = 1. Since the parameter ν is T -periodic, (cid:18) u ( t + T ) u ( t + T ) (cid:19) = Φ( T ) (cid:18) u ( t ) u ( t ) (cid:19) , (49)where the monodromy matrixΦ( T ) = (cid:18) u ( T ) ˙ u ( T ) u ( T ) ˙ u ( T ) (cid:19) . (50)The Floquet coefficients exp( ± µT ) are the eigenvaluesof the matrix Φ( T ), namely the roots of the equation ρ − ρ ∆ + 1 = 0, where the trace of the matrix∆ = u ( T ) + ˙ u ( T ) , (51)while the determinant is unity since the Wronskian oftwo independent solutions is constant. In other words,the coefficients e ± µT = ∆2 ± r ∆ − . (52)The Floquet coefficients are real for | ∆ | ≥ | ∆ | <
2. Substituting Eq. (52) incondition (47) leads to the relation | ∆ | < (cid:18) γT (cid:19) . (53)We can now distinguish different regions.When | ∆ | > γT /
2) the oscillating states donot exist since the function exp( − γt/ Y ( t ) blows upat large time. In the region 2 < | ∆ | < γT / τ R ∼ / [ γ/ − | Re ( µ ) | ] that dependson the value of | ∆ | . While in the region | ∆ | ≤ τ R ∼ /γ . The case | ∆ | = 2 cosh( γT /
2) is physically lessattractive as it not only requires a fine-tuned driving but also results in an asymptotic state with a memoryof the initial conditions.In general it may not be possible to analytically deter-mine the explicit dependence of ∆ on the time-dependentparameters k and γ or more precisely on the function ν of these parameters that appears in the equation (46).Nevertheless we could gain a considerable qualitative un-derstanding of this dependency by probing numerically.To this end we consider the cases where the parametersare restricted to the first harmonics and evaluate ∆ andin turn determine the relaxation times and depict thedomains of the oscillating states.Let us drive the parameters as follows, k ( t ) = k + k cos( ωt ) ,γ ( t ) = γ + γ cos( ωt ) , (54)where k , k , γ , γ and ω = 2 π/T are constants. Notethat the Hill equation can be viewed as an eigenvalueproblem or equivalently as a steady state Schr¨odingerequation of an electron in one-dimensional periodic po-tentials. The parameter k , for instance, can be viewedas an eigenvalue provided the corresponding eigenfunc-tion exist. Unlike the steady state wavefunctions inthe electron case, here the eigenfunctions can explodeasymptotically though not faster than exp( γ t/ k , keeping the other constants fixed, we can ex-pect to encounter both the forbidden regions where theeigenfunctions do not exist and the allowed regions wherethe eigenfunctions and hence the oscillating states exist.Similar features are indeed expected to show up whenthe other parameters are varied too.We find it convenient to plot the modulus of the Flo-quet exponent µ as a function of the parameters in thechosen domain, since ∆ grows exponentially with linearincrease of µ and hence a log scale can be avoided. Thedependence of | Re ( µ ) | for different values of ω on k andon γ is illustrated in Fig. 1(a) and Fig. 1(b) respectively.All the numerical error bars are essentially less than thethickness of the lines used in the plots.As we vary k , one passes through three different re-gions. One of which is where the value of | Re ( µ ) | variesbut is less than γ/ k .The other is where | Re ( µ ) | = 0 and the system can relaxto an oscillating state in a time that is independent of k .The third kind is where | Re ( µ ) | > γ/ | Re ( µ ) | approaches zero for ω = 4 and 6 appear aspoints in Fig. 1(a) only due to the choice of the scale ofthe axis. The time rate of convergence to an oscillatingstate can of course be read from Fig. 1(a) and Fig. 1(b),but is explicitly plotted in Fig. 1(c) to demonstrate thenontrivial dependence of the parameters, say k .It may be required to add the caveat that even thoughfor small k values the relaxation time decreases with in-crease of driving frequency, this is not a generic featurefor arbitrary values of k as is evident from Fig. 1(a). Asimilar conclusion can be arrived at from Fig. 1(b) where FIG. 1. The absolute value of the real part of the Floquet Exponent | Re ( µ ) | controls the stability of the system. If its valueis below the dotted line of γ / γ/
2) in plot (a) and (b), then the system is asymptotically stable. (a) | Re ( µ ) | as functionof k with k = 10, γ = 8, and γ = 4 for frequencies ω ∈ { , , } . (b) | Re ( µ ) | as function of γ with k = 4, k = 2, and γ = 15 for frequencies ω ∈ { , , } . (c) The relaxation time τ − R as a function of k with k = 4, k = 2, and γ = 15for frequencies ω ∈ { , , } . The dotted line denotes the relaxation time when the parameters k and γ are taken to betime-independent ( k = γ = 0).FIG. 2. Stable and unstable region for periodically driven system in (a) k − ω plane with γ = 0, k = 10 . γ = 0 . γ − k plane with γ = 15, ω = 5, and k = 15; (c) k − k plane with γ = 1, γ = 0, and ω = 5. the relaxation time is not always monotonically relatedto the value of γ . Nevertheless observe in Fig. 1(b) thatthe value of γ that saturates the stability of the oscillat-ing state increases with the increase of driving frequency,which is a result that is in tune with the intuition thatperiodic driving enhances stability. Notice what is in-deed far from evident that there is a range of γ valuesfor which γ ( t ) becomes negative at times and yet allowsthe system to be in a stable oscillating state.The non-monotonic behavior of the relaxation timesand the lack of simple algebraic relations to determinestability makes it hard to envisage the exotic landscapeof the parameter space of the driven system containingregions that either further or forbid oscillating states. Wesurvey and chart out some of the planes of the parameterspace, for instance, k − ω plane as shown in Fig. 2(a), γ − k plane as shown in Fig. 2(b), and k − k planeas shown in Fig. 2(c). The maps clearly indicate that aswe move in these planes we could pass through a varietyof terrain starting from vast stretches of stable regionsto tentacles of allowed zones separated by impermissiblegaps. It should be emphasized that the stable regions areidentified not only by ensuring that the condition (53)holds but also by confirming in parallel that the covari-ance matrix (15) is positive definite. B. Concurrence of the perturbations
Suppose we choose the driven parameters k, γ and D such that a normalizable asymptotic distribution in ab-sence of the nonlinear forces exists. We still need to ad-dress whether such a choice is compatible with the con-dition (38) for if it is not the case then the perturbationswill destroy the oscillating state. This condition essen-tially ensures periodicity and boundedness of the pertur-bative corrections to the oscillating state.The difficulty in determining the compatibility is thatthe condition (38) involves Floquet exponents of a Hillequation and hence has to be verified explicitly. We firstrecast this condition to a form similar to Eq. (53) that ismore convenient for distinguishing various regions wherethe perturbative coefficients a Lr exist and are periodic.We shall then proceed to address the issue of compatibil-ity.Let p ( t ) and p ( t ) be two independent solutions of theHill equation (33) with the initial conditions: p (0) = 1, p (0) = 0, ˙ p (0) = 0 and ˙ p (0) = 1. The correspondingmonodromy matrixΦ p ( T ) = (cid:18) p ( T ) ˙ p ( T ) p ( T ) ˙ p ( T ) (cid:19) , (55)and the Floquet coefficients e ± µ p T = ∆ p ± r ∆ p − , (56)where the trace ∆ p = p ( T ) + ˙ p ( T ) . (57)Substituting Eq. (56) in condition (38) leads to the rela-tion | ∆ p | < (cid:18) γ p T (cid:19) , (58)which when satisfied guarantees the coexistence of theperturbations.Since the quantities ∆ p , γ p and ∆ are fixed by the pa-rameters k, γ and D it is natural to ask whether thereare regions in the parameter space where the two rela-tions (53) and (58) are mutually incompatible. Astonish-ingly we found strong numerical evidence to the contraryand noticed the following two relations, γ p = − γ , (59)∆ p = ∆ , (60)for any given k, γ and D that supports an unperturbedoscillating state. We have in fact sampled a class of peri-odic driving that included higher harmonics up to tenthorder and numerically explored a 64 parameter space ei-ther randomly or by continuously varying one of the pa-rameters. Though γ p depends on two other additionalparameters k and D in a nontrivial way, yet we find thatits time average is heedless of their being. Though thefunctions ν and ν p are found to be wildly different fromeach other in general, the corresponding Floquet expo-nents appear to be equal without any exception. Theprime reason for not foreseeing these relations a priori isthat unlike ν the function ν p in addition depends on D not only implicitly through Σ − but also explicitly.We now prove that the relations (59) and (60) indeedhold. The first relation easily follows once we note thatthe dynamics of the determinant | Σ | of the asymptoticcovariant matrix Σ which can be obtained from Eq. (48)can be written as ddt ln | Σ | = − γ + 2 D (Σ − ) . (61)The right hand side of the above equation reducesto − ( γ + γ p ) upon using Eq. (22). The time-period av-erage of the left hand side vanishes since the momentsof the asymptotic distribution are T -periodic, and thusfollows the relation (59).We need a chain of arguments to establish the secondrelation. Using Eqs. (48) it is straightforward to obtainthe equations of motion for all the elements of the inversecovariance matrix S and write down the following systemof nonlinear equations, ddt c = M ( c ) c + d ( c ) (62) where the transpose of c and d ( c ) are c T = (cid:16) S , S , S (cid:17) , d ( c ) T = 2 D (cid:16) S , S S , S (cid:17) , (63)respectively, and the matrix M ( c ) is M ( c ) = k ′ − γ ′ k ′ − γ ′ , (64)where k ′ = k − DS and γ ′ = γ − DS . In thelimit t → ∞ , the matrix S ( t ) approaches the inverseof the asymptotic covariance matrix Σ( t ) and hence k ′ → k p , γ ′ → γ p , the vector d ( c ) becomes T -periodicand the matrix M ( c ) → M ∞ := h − J − + γ p I + J ) + k p J +2 i T . (65)The asymptotic solution of Eq. (62) is unique andbounded since the asymptotic unperturbed distributionis unique and normalizable. Any solution of the equa-tion for a given initial condition approaches the asymp-totic solution, while the differences between the solutionsvanish asymptotically at a rate dictated by the Floquetexponents. The dynamics of the difference δ c of infinites-imally separated solutions follows from Eq. (62) and isgiven by ddt δ c = M ( c ) δ c , (66)which asymptotically takes the exact form as Eq. (27)satisfied by a L for L = 2. Hence, using Eq. (37),we conclude that the Floquet exponents associatedwith the elements of the inverse covariance matrixare µ ( r )2 = γ p + (2 − r ) µ p , where r = 0 , ,
2. The varia-tion of the covariance matrix δS − is related to the vari-ation of the inverse covariance matrix δS by the relation δS − = − S − δSS − → − Σ δS Σ . (67)The Floquet exponents associated with the covariancematrix δS − can be obtained independently and as men-tioned earlier are − γ + (2 − r ) µ . Now using Eqs. (67)and (59), we conclude µ p = ± µ and thus establishEq. (60).Essentially, the two conditions (38) and (47) or the cor-responding equivalent conditions (58) and (53) are notjust compatible with each other but are in fact one andthe same. In other words, we find that the perturba-tions can coexist with the oscillating states in the entiredomain of their existence. IV. CONCLUDING COMMENTS
We have considered a periodically driven Brownianparticle under nonlinear forces and analysed the time0dependence of the asymptotic distribution. The unper-turbed Brownian particle is known to exist in an oscil-lating state under the conditions that the inequality (47)holds and that the covariance matrix is positive definite.To any order in the nonlinear perturbations, we find thatthe oscillating state either can sustain or is destroyed de-pending on whether or not the condition (38) holds. Thereason that it is the same condition that arbitrates theexistence of the oscillating state at every non-zero orderof perturbation is due to the presence of the underlying SL symmetry. We have essentially formulated the per-turbative analysis suitable both for identifying the sym-metry, as given in Eq. (27), and for obtaining exactly theasymptotic distribution in terms of the solution of theHill equation (33). We have obtained the formal expres-sion of the first order coefficients a L ( t ) of the asymptoticdistribution as given in Eqs. (44) and (45), and outlinedthe procedure to determine the higher order coefficients.We have addressed the issue of the stability of the oscil-lating state which is essentially related to the compatibil-ity of the two conditions (47) and (38). These conditionsimplicitly depend on the driving parameters in a nontriv-ial way involving Floquet exponents of the correspondingHill equations. We have charted out some of the terrainsof the oscillating states in the parameter space of drivinginvolving first harmonics so as to explicitly demonstrate the nontrivial relation between driving parameters andthe stated conditions. More importantly, we have provedthe equivalence of these conditions inspite of their im-plicit dependence on the parameters and established thatthe oscillating states are stable against nonlinear pertur-bations to all orders of perturbation.The ubiquity of the driven systems and their accessto experiments is a strong motivation to study the ef-fect of different perturbations on the oscillating states.The perturbative formulation developed here could provevaluable not only in understanding the properties of thesestates but also in establishing the appropriate descriptionof driven physical systems.Since a variety of macroscopic systems undergostochastic dynamics similar to that of Brownian motion,these systems when driven has the possibility to exist instable oscillating states. 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