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Fluctuation-dissipation relations for steady state systems

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Fluctuation-dissipation relations for steady state systems
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    a  r   X   i  v  :   0   9   0   8 .   4   5   4   1  v   2   [  c  o  n   d  -  m  a   t .  s   t  a   t  -  m  e  c   h   ]   1   7   M  a  r   2   0   1   0 Fluctuation-dissipation relationsfar from equilibrium Jianhua XingDepartment of biological sciences, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061 Abstract The fluctuation-dissipation (F-D) theorem is a fundamental result forsystems near thermodynamic equilibrium, and justifies studies betweenmicroscopic and macroscopic properties. It states that the nonequilibriumrelaxation dynamics is related to the spontaneous fluctuation at equilib-rium. Most processes in Nature are out of equilibrium, for which we havelimited theory. Common wisdom believes the F-D theorem is violated ingeneral for systems far from equilibrium. Recently we show that dynam-ics of a dissipative system described by stochastic differential equationscan be mapped to that of a thermostated Hamiltonian system, with anonequilibrium steady state of the former corresponding to the equilib-rium state of the latter. Her we derived the corresponding F-D theorem,and tested with several examples. We suggest further studies exploitingthe analogy between a general dissipative system appearing in variousscience branches and a Hamiltonian system. Especially we discussed theimplications of this work on biological network studies.  It is ubiquitous to observe a system at a state invariant with time (with theapproximation that the relevant constraining parameters changes much slowerthan the time scale under interest). It can be a thermodynamic equilibriumstate, or more likely a nonequilibrium steady-state. Some examples are homeo-static states of living organisms, a stable eco- or financial system. Quite often itis important to know how a system initially subject to a perturbation relaxes toa steady state (including the equilibrium state) after removal of the perturba-tion. The fluctuation-dissipation theorem states that the relaxation dynamicsfor a process close to equilibrium is related to the spontaneous fluctuation atequilibrium. Originally formulated by Nyquist in 1928[1], and first proved byCallen and Welton in 1951[2], the F-D theorem is related to many importantresults in statistical physics. Examples are the Einstein-Smoluchowski relationon the diffusion constant and drag coefficient [3, 4], Onsager’s regression hy-pothesis[5,6], and the linear response theory[7]. The F-D relation also has practical importance. It allows deducing nonequilibrium dynamics from equi-librium measurements, and justifies the relation between macroscopic dynamicsand microscopic level simulations, e.g., calculating the diffusion constant. Inrecent years, fluctuation theories of nonquilibrium processes, especially for sys-tems far from equilibrium received great attention[8, 9,10,11,12]. On studying problems from physics, chemistry, cellular biology, ecology, en-gineering, finance, and many other fields, the following form of equations arewidely used[14, 15, 16], dx i /dt = G i ( x ) + m  j =1 g ij ( x ) ζ  j ( t ) ,i = 1 , ··· ,n. (1)In general m and n may be different, ζ  j ( t ) are temporally uncorrelated, statisti-cally independent Gaussianwhite noise with the averagessatisfying < ζ  j ( t ) ζ  ′ j ( t ) > = δ jj ′ δ ( t − t ), g ( x ) is related to the n × n diffusion matrix gg T  = 2 D /β , where thesuperscript T  refers to transpose. For a physical system β is the inverse of theBoltzmann’s constant multiplying temperature. For a non-physical system, onecan define an effective temperature relating to β . Recently we proved that thereexists a mapping between a stochastic dissipative system described by Eqn.1and a thermostated Hamiltonian system[17]. The mapping allows many re-sults from equilibrium statistical physics directly applicable to nonequilibriumprocesses. Specifically in this work, we will derive the F-D relation applica-ble to processes far from equilibrium. An F-D relation, if exists, would allowpredicting the relaxation dynamics to a steady-state through measurements of steady-state fluctuations. The latter are in general easier to measure.1  1 Theory 1.1 Existence of mapping between linear stochastic dissi-pative systems and Hamiltonian systems First we briefly summarize the main result of [17]. The mapping is based aseminal work of Ao, which shows that one can always construct a symmetricmatrix S and an anti-symmetric one T , and transform Eqn.1into, [18] ( S + T ) d x dt = ( S + T )( G ( x ) + g ( x ) ζ  ( t ))= −∇ x φ ( x ) + g ′ ( x ) ζ  ( t ) (2)where φ is a scalar function corresponding to the potential function in a Hamil-tonian system satisfying ( ∂  × ∂φ ) ij ≡ ( ∂  i ∂  j − ∂  j ∂  i ) φ = 0, and g ′ g ′ T  = 2 S /β .Then S and T are uniquely determined by ∂  × [( MG ( x )] = 0 , ( M ) − 1 + ( M ) − T  = 2 gg T  , (3)where M = S + T , with proper choice of the boundary conditions. In [17], wefirst demonstrated that starting with the corresponding Fokker-Planck equa-tions, one can derive the transformation matrix more transparently following astandard procedure used by Graham and by Eyink et al. previously [19, 20].Then we showed that one can map the dynamics described by Eqn.2to aHamiltonian system in the zero mass limit, H  =( ˜p − A ( x )) 2 2 m + φ ( x )+ N  α  α =1  N   j =1  12  p 2 αj +12 ω 2 αj ( q αj − a α ( x ) / ( √ Nω 2 αj )) 2  (4)The term A is a vector potential satisfying T  ij =  j  ∂A i ∂x j − ∂A j ∂x i  . The lastterm is a type of bath Hamiltonian discussed by Zwanzig[21]. The Hamilto-nian mathematically corresponds to Dirac’s constrained Hamiltonian [22]. Itdescribes a massless particle, coupled to a set of harmonic oscillators, mov-ing in a hypothetical n -dimensional conservative scalar potential and magnetic(the vector potential) field. The steady state distribution is thus given by theBoltzmann distribution of the Hamiltonian system, ρ ss ( x ) =   d p d Y exp( − βH  )   d x d p d Y exp( − βH  )=exp( − βφ )   d x exp( − βφ )(5)where Y represents all the bath variables. Eqn.5is also conjectured by Ao[18], and a general proof is given in[23]. 2  1.2 General fluctuation-dissipation theorem Let’s denote the steady-state ensemble average of a generic dynamic quantity O as, < O > =   d xO ρ ss ( x ) (6)Consider a system initially at a steady state defined by Eqn.1with an ex-tra infinitesimal perturbation δ G ( λ ), where λ refers to the parameters be-ing perturbed. The corresponding perturbation Hamiltonian term is given by ∇ ( δH  ) = − ( M ′ G ′ − MG ) ≈− M δ G − δ MG , where δ M is the variation of  M due to the perturbation. At time 0 δ G is removed, and the system relaxes tothe steady-state of  G . Or the Hamiltonian of the corresponding mapped systemchanges from H  ′ = H  + δH  = H  + ∂φ∂λ · δλ . Then the nonequilibrium relaxationdynamics of  O follows¯ O ( t ) − < O > =   d x d pO ( t )exp( − βH  ′ )   d x d p exp( − βH  ′ ) − < O > ≈− β [ < O ( t ) ∂φ ( x (0)) ∂λ ) > − < O ><∂φ∂λ> ] · δλ (0)(7)In the above derivation, we used the stationary property of equilibrium ensembleaverage, so < O ( t ) > = < O (0) > = < O > . This is the generalized F-D relationfor systems obeying or violating detailed balance, which states that nonequilib-rium relaxation dynamics can be predicted from steady-state fluctuations. If one relates the relaxation function to the linear response function,¯ O i ( t ) − < O i > ≈   tt 0 dt ′ χ ij ( t − t ′ ) δλ j ( t ′ ) (8)one obtains the differential form of the F-D relations, χ ij ( t − t ′ ) = − βddt  < O i ( t ) ∂φ ( x ( t ′ )) ∂λ j ) >  (9)Eqns7and9are the central results of this work. These results are actually mathematically trivial with the replacement φ = − ln ρ ss [12]. The mapping,however, provides direct connection between ρ ss and the Langevin equations(see especially case 4 in the next section), and allows unified treatment forsystems obeying or violating detailed balance. One can generalize the resultsdiscussed in this work to higher orders of  δ G .One type of perturbation of special interest is a system coordinate cou-ples to some constant external force linearly, δH  = f  · x , which corresponds to δ G = − M − 1 f  with δ M = 0. Notice that δ G is in general a nonlinear func-tion of  x . This situation has been previously discussed by Graham, and byEyink et al. [19,20]. Under this special type of perturbation, all the famil- iar results obtained on studying relaxations near an equilibrium state follow3  [24]. One can define a response function χ ( t,t ′ ) so¯ x ( t ) − < x > =   ∞−∞ dt ′ χ ( t − t ′ ) f  ( t ′ )+ O ( || f  || 2 ). The function χ ( t − t ′ ) is stationary and satisfies the Kramers-Kr¨onig relations. If  f  is time varying with a monochromic frequency, f  = Re [ f  ω exp( iωt ) + f  ∗ ω exp( − iωt )], the system absorbs ”energy”, with the absorp-tion spectrum abs ( ω ) ∝ ω 2   ∞ 0 dt f  T ω < ( x t − < x > ( x 0 − < x > ) T  > f  ∗ ω cos( ωt )[24]. 2 Special cases and numerical tests Here we will show that several versions of the previously derived generalizedF-D relations are special cases of the above results. We will also consider sev-eral pedagogical examples, especially biochemical networks, to demonstrate thevalidity of our results. Case 1 : The first one is an analytically solvable irreversible linear chemicalnetwork (Fig. 1a). The system is initially at a state with k 0 + δk . Then at time 0it changes to k 0 , i.e. , the inflow flux varies. The system relaxes to a new steady-state. The perturbation is δ G = ( δk, 0 , 0), and the corresponding Hamiltonianperturbation term is δH  = − ( x 1 ,x 2 ,x 3 ) M δ G . The dynamic equations are, ddt  x 1 x 2 x 3  =  − 2 0 01 − 1 11 0 − 1  x 1 x 2 x 3  +  k 0 00  +  zζ  1 ( t ) zζ  2 ( t ) zζ  3 ( t )  (10)For simplicity of the following discussions, we write the above equation in theform d x /dt = Kx + b + zζ  ( t ). The predicted relaxation function through thefluctuation-dissipation relation is given by,(¯ x i ( t ) − < x i > ) FD ∝ C · M · δ G (11)where ( C  ij ( t )) = < ( x i ( t ) − < x i > )( x j (0) − < x j > ) > . We also tested anincomplete relaxation function by setting M as an identity matrix,(¯ x i ( t ) − < x i > ) incomplete ∝ C · δ G (12)Analytical solutions are given in the appendix. Fig. 1 shows that ∆ x i ( t ) calcu-lated from the F-D relation (Eqn11) reproduces the exact result given by Eqn21(solid line). However, without the term M , the incomplete F-D relation Eqn12predicts an initial increase of  x 2 (and similar for x 3 ). Physically, sponta-neous fluctuation of  x 1 is anti-correlated with those of  x 2 and x 3 concentrationsthrough reactions 1 and 3. The incomplete F-D relation erroneously attributesthis mechanism to the relaxation dynamics.In the second example, we still use the network shown in Fig. 1a, but assumethat all the reactionsexcept reaction0 followthe Michaelis-Mentenkinetics, e.g. , dx 1 dt = k 0 − v 1 x 1 K  1 + x 1 − v 3 x 1 K  3 + x 1 + zζ  1 ( t ) (13)4
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