Master Langevin equations : Origin of asymptotic diffusion

We extend the master-equation treatment of dynamical evolution of a system-plus-reservoir configuration including the propagation of initial correlations as a noise source. Specializing into the quantum harmonic oscillator coupled to a fermionic heat bath, we develop a model for the diffusion matrix in the space of diagonal density operators. It can be shown that mean values of observables undergo Langevin-like motion and, in particular, that the mean value and dispersion of the oscillator quanta approach the canonical equilibrium values. A final interpretation of the characteristics and role of the noise source is given.


I. INTRODUCTION
The irreversible dynamics of two coupled components of a macroscopic, isolated system is a well-known text- book subject in statistical physics [1].The most popular and useful realizations of such a situation are the Brown- ian particl" "ventually subjected to a conservative ex- ternal force fieldin its thermal reservoir and the sin- gle particle in an environment of identical partners, to which it couples through a two-body interaction.The standard probabilistic approaches lead one to formulate the above problem in terms of the density matrix or the distribution function of the selected particle, whose evo- lution is governed by a master equation in the case of the Brownian object, and by a kinetic equation for the N body system [2,3].Accordingly, the expectation values of the observables of interest satisfy damped equations of motion, whose solutions exhibit asymptotic decay to- wards the thermodynamic equilibrium values.Comple- mentary to the dissipative behavior of the mean values, the covariances and dispersions bring into evidence the diffusive action of the heat bath, whose strength is related to the intensity of the damping through a Huctuationdissipation relationship [4 -7].Such effects have been extensively investigated for both the classical and the quantal Brownian motion; however, a statement of the Huctuation-dissipation theorem in the quantal case has not been made for general thermal environments sur- rounding the macroscopic particle.
More recently, due to the increasing amount of ex- perimental work devoted to extract information of far- from-equilibrium dynamics in N-fermion systems such as nuclei, new insights have been put forward regarding the role of the reservoir.The initial cluster correlations [8] seem to be very substantial in the process of fragment formation preceding the disassembly of a hot nu- cleus, and the so-called phase-space Huctuations [9,10] are capable of accounting for a variety of dynamical ef- fects involving the moments of the one-body distribution function.In this context, and in the same spirit of for- mer authors such as Bixon and Zwanzig [11] and Van  Kampen [12], Ayik and Gregoire [9] have recently devel- oped an extension of the Boltzmann-Uehling-Uhlenbeck (BUU) kinetic equation that attempts to incorporate the initial two-body correlations with which the N-body system is constructed, in the manner of a fluctuating source appearing in the above equation of motion.The dis- tribution function in the Wigner representation then becomes a stochastic variable in distribution space, and this character is transmitted to every mean value of dynamic observables.This view has been complemented by the approach of Randrup and Remaud [10], who derive the same evolution equations in phase space stemming from a Fokker-Planck description of the stochastic Wigner func- tion.While several numerical experiments have been al- ready published [13, 14] along these lines, the manifesta- tions of the noise upon the collective variables have been also investigated [15] in the close-to-equilibrium regime, giving rise to a clear Brownian behavior with a well- identified fluctuation-dissipation relationship.
In the standard derivation of the master equation, the initial system-reservoir correlations are usually dis- regarded; such a procedure is valid insofar as one is in- terested in the asymptotic regime, since usually the life- time of microscopic correlations is short in a macroscopic environment.However, as in the N-particle system, ini- tial correlations may be relevant in the small time scale, where their propagation gives rise to stochastic kicks on the evolving density matrix of the Brownian particle.It is then of interest to take a deeper view on the characteris- tics of the noise associated to the initial system-reservoir correlations and attempt to achieve an interpretation of the origin of their fluctuation-dissipation relationship.For this sake, in this work we adopt Ayik and Gregoire's point of view and develop a model for the fluc- tuating source in the master equation of a quantal har- monic oscillator in a fermionic heat bath [16 -19].This is presented in Sec.II.In Sec.III, we discuss the effects of this noise on the oscillator observables and dispersions; in particular, we give a prescription to extract the diffusion coefflcient for any quantity, once the diffusion matrix in probability space is known.The whole procedure is il- lustrated computing the correct dispersion in the phonon number.The discussion and Anal summary is presented in Sec.IV.

II. STOCHASTIC BROW'NIAN MOTION MODEL
The physical problem under consideration is the relax- ation of a sytem 8 due to its coupling to a heat reservoir 72,.Let H be the total Hamiltonian of the combined system-plus-reservoir configuration, H = Hg+ HR+ Hg~, with HsR the driving interaction and let, at any time t, where bR"(t is the contribution of the noise source to the time derivative of the population .A more elaborate description of the fluctuating kernel (5) depends on the specific choice of the interaction HsR, which in turn determines the structure of the downwards and upwards transition rates, respectively, W+ and W in Eq. ( 4).In favor of a defi- nite illustration, we select the quantal Brownian motion (QBM) [16 -19] model that considers the coupling of the oscillator to a ferrnionic heat bath.The QBM interaction be the total density operator, with psR the correlated, i.e. , nonfactorizable, contribution to the statistical rna- trix.The well-known reduction-projection techniques [16] lead to the master equation for the reduced density matrix of the system in the form pS(t) = i [HS + Tr-RHSRpR(t), pS(t)] where I t and bt (I' and b) are boson and fermion cre- ation (annihilation) operators, respectively.The transi- tion rates read as where k, , e, , and p, are the momentum, the single parti- cle, and the Fermi-Dirac population of the fermion state ~i), and 0 and q are the energy and momentum of an oscillator quantum, related by the dispersion law where Hg~denotes the interaction Hamiltonian in Heisenberg representation and TrR indicates a tracing operation on the heat bath variables .A symmetric equa- tion holds for the thermal reservoir; however, in most ap- plications the latter is an extended object assumed to be in thermodynamical equilibrium at any stage of the time evolution of the system 8, i.e. pR(t) = pR(0) = pR.
In commonly investigated situations appearing in quantum optics and nuclear or condensed matter physics, the last term on the right-hand side of Eq. (3) plays no role, either because one assumes that no initial correlations have been built in the total system, or due to the additional hypothesis that, even if such correlations had been present, they would have damped away within a time scale much shorter than the actual observation time t.It is then clear that within the lifetime associated to the evolution kernel e ' ', the macroscopically large di- versity of choices for the irreducible matrix p&'R, which remains an unknown entity for an observer concentrated on the system S, permits one to regard the propagating correlations as some external stochastic noise.In this sense, the reduced density ps(t) is a stochastic process in Liouville space that undergoes Brownian motion ac- cording to the functional Langevin equation (3).
In order to Bx ideas, we consider a quantum harmonic oscillator coupled to an arbitrary heat reservoir.The master equation for the occupation probability p"of the nth oscillator state is well known; thus in the Markovian limit one can write A = c, fq/ with c, the speed of sound in the macroscopic environ- ment.Furthermore, one has the detailed balance rela- tionship (9) at any temperature T; the transition rates (7) can be explicitly computed for a free Fermi gas [20], assuming constant coupling matrix elements A~"= A, giving m ~A~~1+ exp[ -P(2mcW In these expressions, Az = (2m/AT)~is the thermal wavelength of fermions of mass m, ni is the mean number of oscillator quanta at temperature T, and p is the fermion chemical potential.Moreover, in- troducing the effective decay rate p"(t) = W [(n+1)p"~i -np"] +W [np", -(n~1)p"]+a@"(t), (4) we realize that W+ -v(l + nr), R' = Vn1.Now considering the interaction (6), after some algebra one finds the following simple appearance for the noise source in Eq. ( 5): As in previous developments of the QBM model [16 -20], the notation here employed for matrix elements indicates a transition from a configuration with a given number of oscillator quanta and one labeled fermion state, to an- other configuration with plus or minus one boson and a difFerent fermion state, the remaining N -1 fermions not undergoing any transition.If we now introduce an ensem- ble of initial conditions on which all fluctuating quanti- ties may be averaged, it is clear that the noise source (5) averages to zero, i.e. , bK"(t) = 0, Vn, since the ensemble contains all possible matrix elements of p&'R appearing in ( 16) with all possible phases.One needs to know the correlation 6K"(t)bK"', (t) in order to characterize the fluctuations.A computation of such a quantity for a quantum system has been performed by Ayik and Gregoire [9] in the frame of the Boltzmann- I angevin equations for fermions [actually, the so-called (BUU) kinetic equation enriched with the propagation of initial two-body correlations].Their derivation is based on two major assumptions: (i) the two-body correlation operator p2 is expressed as the Hermitian adjoint of the two-body interaction; and (ii) the time correlation of pz at time t = 0 can be replaced by its value at a finite time I bK (t) = -& ).
[& v n{n -1 ~l ps' Inp) -A*"v'n+ l{n, o ipse'~~n+ lp)]e '( '-+'~)' A test of the validity of this assumption is the fact that this correlation kernel gives the appropriate diffu- sion coefBcients for one-body, momentum dependent observables [15].In what follows, we briefly summarize the main steps and hypothesis of their procedure as applied to the present case, for which we interpret the correlation matrix elements in Eq. ( 16) as the transition amplitudes = {Cs~(n)il' b"b I'b, b"~Cs~(n')).
The indicated matrix element then becomes a product of boson and fermion occupation numbers.In addition, in the noise correlation, each term contains a phase factor, which for the above contribution is e'(+ (2) The noise correlation thus obtained is now inte- grated over the time difference tt .This gives rise to the energy conserving kernel b(A -u ") that appears in the transition probabilities in Eq. ( 7).
(1) As one writes the noise correlation, i.e. , 6K"(t)b'K', (t) in terms of (16), eight terms involving a product of two correlation matrices appear.One then assumes that the averaging procedure in the ensemble of initial conditions, denoted by the overbar, suppresses the intermediate generalized projector operator in the product p~')(t)p('l(t').In other words, a typical contribution -r, n(W+p"+ W-p"r) -b +r, (n + 1)(W+p +r + W p)). - In this expression, p is the averaged mth level popula tion at time t, , namely the solution of the deterministic master equation [21], This reflects, on the one hand, the fact that the fluc- tuating kick at time t originates in a correlation propa- gated from some arbitrary previous instant.On the other hand, the diagonal term of D""exhibits the gain-plus- loss structure characteristic of difFusion processes [12], a structure that shows up as well in the sum of both off-diagonal terms.In particular, one may compute the difFusion matrix (20) at equilibrium; let p~= e" The role of D""(t)as the diffusion matrix for the popu- lation of oscillator quanta becomes clear on very simple grounds.
The boson occupation number nj is then recovered as the equilibrium ensemble averaged first moment; its co- variance (33), (36) is Deq oz, (t) = ' (1 -e "'), (44) while a straightforward calculation of the diffusion coef- ficient defined in Eq. (23) gives, at any time, Since the analytical solution of the deterministic system for the ensemble averages ng is known, one might in principle fold the fluctuations (41) into that solution and obtain the fluctuating moments nt, (t) .A set of correla- tions o&~&, --bnkbng~could be established in this manner; this is just an alternative to the covariance matrix o. of Eq. ( 27).Being both a moment and an observable, the mean number of quanta nq, proportional to the mean ex- citation energy of the oscillator, is specially interesting.One easily obtains from (40), 6q(t) = q(t) -q(t) =).Q 6p  Equation (47) expresses a well-known relationship for the dispersion of the mean photon number in a fluctu- ating electromagnetic field with detailed balance (see, for example, Refs.[22,23] and references therein), while Eq. ( 46) is the particular fluctuation-dissipation relation- ship for a boson system.The specific nature of the driv- ing mechanism and the thermal environment appear in the dissipation rate v, which for a fermionic reservoir reads [cf.Eqs. ( 10) and ( 13)] as 1+exp[ -P(2rnc, + s' q -20 -p)] q A2~1+ exp[ -P(-, 'rnc2+ s' q'+ 2iA -p)] (48)

IV. SUMMARY AND CONCLUSIONS
In this work we have analyzed the temporal evolution of the QBM model (i.e. , a fermionic system coupled to a bosonic heat reservoir) when the eff'ects of the initial cor- relations between the system of interest and the bosonic bath are taken into account.
These correlation terms give rise to a noise source whose statistical properties are given by its mean value (over the ensemble) and its temporal correlation.The latter has a structure of the type gain+loss which is char- acteristic of the diffusion process.This term is of primary importance because it appears in the evaluation of the fluctuations of the observables.
At this point, it is interesting to summarize the way in which this term is calculated.First, using the stan- dard techniques developed for the resolution of the QBM model one obtains an expression for the time correla- tion of the noise source .This result already contains a specific selection of the system-plus-reservoir correla-tion operator.Next, one sums up all the contributions for all times and then one makes the assumption intro- duced in Ref. [9] that at any time t the correlation of the noise source can be replaced by an efFective white Gaussian one with the same diffusion matrix.This is the expression that we use to calculate the fluctuations of the observables; in particular, when the observable is the occupation number ni the correct asymptotic limit is obtained.
It must be stressed that the above procedure is in fact equivalent to replacing the true initial (nonkinetic [2]) correlation by some kinetic one which gives the same diffusion matrix in density matrix space.In this way one does not keep track of uncontrollable correlations; instead, one contemplates in the description only the strictly kinetic ones appearing in the collision kernel cor- responding to the close to equilibrium evolution, which of course produce the correct asymptotic regime.This means that although some extra degree of stochasticity has been introduced in the description, it is not related to the early history of the system but is a manifestation of the dynamics itself in the long-time run.
The results here encountered support the idea that the kinetic correlations are those responsible for the asymp- totic diffusive behavior of real systems, once the initial nonkinetic ones have been washed away by the collisions.