Two-temperature Langevin dynamics in a parabolic potential
We study a planar two-temperature diffusion of a Brownian particle in a parabolic potential. The diffusion process is defined in terms of two Langevin equations with two different effective temperatures in the and the directions. In the stationary regime the system is described by a non-trivial particle position distribution , which we determine explicitly. We show that this distribution corresponds to a non-equilibrium stationary state, characterised by the presence of space-dependent particle currents which exhibit a non-zero rotor. Theoretical results are confirmed by the numerical simulations.
, , ,
Keywords: Two-dimensional diffusion, parabolic potential, non-equilibrium stationary state, rotating flows
The idea of physical systems characterised by two different temperatures has been proposed a long time ago for the models of spin-glasses and neural networks with partially annealed disorder [1, 2, 3, 4]. In these models, the two temperatures and are related to two different degrees of freedom, which are evolving at two essentially different time scales. As an example, one may consider a system in which the fast spin variables are connected with the thermal bath kept at the temperature , while the slow spin-spin coupling parameters are connected with another thermal bath maintained at the temperature . It can be easily shown that in the stationary (non-equilibrium) state the statistical properties of such systems are described by the usual replica theory of disordered systems with a finite value of the replica parameter (see also ). Unfortunately, generalisation of this idea to the case when dynamics of two types of degrees of freedom is characterised by two comparable (or equal) time scales turned out to be rather problematic: it seems that there is no generic explicit expression for the stationary probability distribution function which would generalise the Gibbs distribution of the equilibrium case . However, there is a particular case for which one can find an explicit and a rather non-trivial expression for the stationary distribution function. Namely, this is the case when the two degrees of freedom and related to the thermal baths with the temperatures respectively, experience a potential which is a quadratic function of and [6, 7]. During last decade theoretical investigations of such type of systems were mostly concentrated on the studies of nonequilibrium fluctuations and energy transfer . Recently this type of model was studied both theoretically  and experimentally  from the point of view of the entropy production and memory effects. In this paper, keeping in mind putative experimental realisation of such a type of systems, we are going to discuss the two-temperatures situation reformulated in terms of the two-dimensional diffusion of a Brownian particle in a parabolic potential. The diffusion process is defined in terms of Langevin dynamics with two different effective temperatures in the and the directions. In the stationary state this system is described by a non-trivial distribution function , which can be computed explicitly. Unlike for the equilibrium case (), this non-equilibrium stationary state is characterised by the presence of nontrivial space dependent particle’s flows . Moreover, these flows exhibit a ”symmetry breaking” rotor, (directed perpendicular to the -plain), the sign (or the direction) of which is determined by the temperature difference .
The paper is organised as follows. In Section 2 we define our model and present the explicit solution for the stationary particle’s probability distribution function . In Section 3 we compute putative ”observable” quantities of the system, such as the variances of the particle displacements in the and the directions, the rotor of the particle’s flows as well as the average rotation velocity. In Section 4 we report the results of the numerical simulations and compare them with our analytical predictions. Finally, in Section 5 we conclude with a brief recapitulation of our results.
2 The model
We consider stochastic, over-damped Langevin dynamics of a particle moving in a two-dimensional space in a presence of an external potential . The particle instantaneous position is defined by projections on the and the axes, and , respectively. The time evolution of and is described by following equations:
Here is anisotropic stochastic noise, with zero mean and correlation function
where and are two different ”temperatures” and has the following parabolic form :
The shape of the potential is controlled by the parameter . To keep the particle localised near the origin, we have to impose the constraint . This follows from the requirement that both eigenvalues of the potential, , must be positive; in the case , there is a direction in the plane at which the potential has a negative curvature which allows the particle to escape to infinity.
In the stationary regime, the probability distribution function of the particle position obeys the stationary Fokker-Planck equation :
In the trivial isotropic case, , the solution of the above equation is simply the equilibrium Gibbs distribution .
One can easily show that in the generic anisotropic case with arbitrary and , the solution of the stationary equation (4) reads:
where the following shortenings have been used
Further on, is the normalisation constant (the ”partition function”), defined as
One immediately observes that exists, so that the system has the stationary solution, only for .
3 The observable quantities
3.1 Variances of particle positions
Using the above probability distribution function we can straightforwardly calculate the variances of the particles position with respect to the and the axes:
The characteristic quantity, which can serve as the measure of anisotropy in the system under study, is defined as the ratio of these two quantities :
In the trivial decoupled case, , we find , while in the isotropic case, we have for all values of the coupling parameter . Note next that in the strongly anisotropic case, e.g., when , one has
In other words, in the strongly anisotropic case the values of both and are defined by the largest , while the value of the ratio becomes a -independent constant.
3.2 Mean rotation velocity
In the stationary case the current is defined as follows:
Note that in the isotropic case, , we have , so that . In the anisotropic case the above non-trivial pattern of currents can be characterised in terms of the rotor:
In general, the rotor is a rather complicated function of two variables and , but it is remarkable that the function has a non-zero (and very simple) value at the origin at :
Note that this quantity changes sign from minus (”left rotation”) at , to plus (”right rotation”) at .
Due to the presence of a non-zero particle’s current rotor, one finds that the mean particle’s rotation velocity is also non-zero. Indeed, for a given value of the particle’s linear velocity located in the point on the two-dimensional plane, its angular velocity is
where is the vector product directed along the -axis. Thus, the mean rotation velocity in the limit of an infinite observation time can be defined as follows:
Changing averaging over time by averaging over ensemble (which will be justified in what follows by numerical simulations) we get:
where the parameter is defined in eq.(9).
One can easily prove that the maximal value of the mean angular velocity is , and it is achieved either in the limits (which corresponds to for finite ) or in the limit (which corresponds to for a finite ), and the value of the coupling parameter .
4 Numerical simulations: Brownian dynamics
To verify our analytical predictions and the underlying assumption that the time-average can be replaced by the ensemble average, we perform numerical simulations of appropriately discretised Langevin equations eqs.(1). Substituting the potential into eqs.(1) we first write these equations explicitly:
where the variances of the thermal noise components are defined by , and .
Discretising Eq. (30) with a time step , we have :
where and are delta-correlated random numbers with Gaussian distribution of unit half-width, , , which are the conditions of a smooth motion. In that case for a free motion of a particle () which starts at the origin (), the diffusion coefficients are , and the variances of the displacement are given by and . In the case of the symmetric potential (), one has in the stationary regime and , independently of . For asymmetric potential , we will compute the mean angular velocity given in eq.(24) and the measure of anisotropy that is described by eq.(13).
The numerical simulation has been done for the time step . The averaging has been performed over the total time period time units, the numerical inaccuracy has been evaluated by splitting the whole time interval into sub-intervals. In Fig. 1(a),(b) we plot numerical results for the ratio of variances and for the mean angular velocity , calculated as the time-average of , as functions of for , . For comparison we also show our analytical predictions in eq.(13) and eq.(24), respectively, and find a perfect agreement. This justifies the replacement of the time-average by the ensemble average in our analytical calculations.
Further on, in Fig. 2(a),(b) we plot the same quantities as functions of (with ) for . We again observe a very good agreement between our numerical and analytical results.
In the present work we studied a simple stochastic ”toy model” with only two degrees of freedom which are connected to two thermostats maintained at two different temperatures and , respectively. The model describes diffusion of a particle on a two-dimensional plane in a presence of a parabolic potential such that the stochastic noises in the and the directions have different strength ( and , respectively). We determine the stationary state probability distribution function for the position of the particle. Despite its relatively simple structure, it turns out to be rather non-trivial, revealing interesting qualitative physical phenomena. In particular, in the stationary state one finds a rather sophisticated pattern of particles’ density currents (which would be identically equal to zero in the equilibrium case) characterised by the non-zero rotor. Moreover, due to the presence of this flux rotor one observes the phenomenon which could be interpreted as a ”spontaneous symmetry breaking”, namely one finds non-zero value for the average particle’s rotation (around the origin) velocity. This value is proportional to , eq.(29), being positive (left rotation) for and negative (right rotation) for .
It should be stressed, however, that except for recently proposed two-temperature electric analog system , for the moment the considered model has no experimental realization. Thus, the aim of the present work is somewhat provocative: we would like argue that the systems of such type are sufficiently interesting to stimulate investigations for their ”hardware” implementations. We also believe that modifications of our toy model towards a system that could be realised in practice and at the same time would not loose its interesting behavior (rotation), is possible.
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