Full absorption statistics of diffusing particles with exclusion
Suppose that an infinite lattice gas of constant density , whose dynamics are described by the symmetric simple exclusion process, is brought in contact with a spherical absorber of radius . Employing the macroscopic fluctuation theory and assuming the additivity principle, we evaluate the probability distribution that particles are absorbed during a long time . The limit of corresponds to the survival problem, whereas describes the opposite extreme. Here is the average number of absorbed particles (in three dimensions), and is the gas diffusivity. For the exclusion effects are negligible, and can be approximated, for not too large , by the Poisson distribution with mean . For finite , is non-Poissonian. We show that at . At sufficiently large and the most likely density profile of the gas, conditional on the absorption of particles, is non-monotonic in space. We also establish a close connection between this problem and that of statistics of current in finite open systems.
Keywords: non-equilibrium processes, large deviations in non-equilibrium systems, stochastic particle dynamics (theory), current fluctuations.
Statistics of large fluctuations of current in non-equilibrium steady states of diffusive lattice gases has become a central topic of non-equilibrium statistical mechanics Prahofer (); Jona05 (); Jona06 (); MFTreview (); Bodineau2004 (); DDR (); Derrida07 (); Harris (); Prolhac (); Maes (); Lecomte2010 (); Gorrisen (); Akkermans (); HEPG (). The “standard model” here involves a lattice gas between two heat baths kept at different temperatures, or between two reservoirs of particles at different densities. Most of the works on this subject assumed one-dimensional geometry. It is interesting to see what new effects high dimensions can bring Akkermans (); vortices (). Here we consider a simple setting which can be studied in arbitrary dimension. Suppose an infinite lattice gas of density is brought in contact with an immobile macroscopic (for simplicity, spherical) absorber. The gas particles are absorbed immediately when they hit the absorber. Here there is only one reservoir: the absorber which enforces a zero gas density in its vicinity. This simple setting has a long history. It was originally suggested by Smoluchowski S16 () as a minimalistic model of diffusion-controlled binary chemical reactions (where the absorber mimics a very large particle of the minority species). The average particle flux into the absorber mimics the reaction rate C43 (); R85 (); OTB89 (); Havlin (). Here we are interested in large fluctuations of the particle flux, and the two main questions we ask are the following:
What is the probability distribution that particles are absorbed during a very long time ? (The long-time limit is achieved when becomes much greater than the characteristic diffusion time determined by the absorber radius and the gas diffusivity.)
What is the most probable density history of the gas, conditional on the absorption of particles during the time ?
The special case of (all the gas particles survive until time ), corresponds to the celebrated survival problem. This problem, and its extensions, have been extensively studied in the past Donsker (); ZKB83 (); T83 (); RK84 (); BZK84 (); BKZ86 (); BO87 (); Oshanin (); BB03 (); Carlos (); Havlin (); KRbN (); BMS13 (). Most of these studies assumed that the gas is composed of non-interacting Brownian particles in a continuous space, or non-interacting random walkers (RWs) on a lattice. An account of interactions between the particles (which is important, for example, in crowded environments such as a living cell crowd ()) makes the problem much harder. Recently, the survival problem with interactions has been addressed in Ref. MVK () for diffusive lattice gases. In the hydrodynamic limit, the coarse-grained density of these gases is governed by the diffusion equation
where is the diffusivity. Large-scale fluctuations in these gases are described by the Langevin equation
where is a zero-mean Gaussian noise, delta-correlated in space and in time Spohn (). As one can see, the coarse-grained description of the fluctuations includes an additional transport coefficient, . This coefficient comes from the shot noise of the microscopic model, and it is equal to twice the mobility of the gas Spohn ().
Here we considerably extend upon the previous work by investigating the full absorption statistics of particles in diffusive lattice gases, and by providing answers to the two main questions formulated above. The long-time absorption statistics can be conveniently described by the macroscopic fluctuation theory (MFT) of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim. The MFT is a variant of WKB approximation applied to Eq. (2), see Ref. MFTreview () for a recent review. Employing the MFT, the authors of Ref. MVK () studied the survival probability and the optimal (most likely) density history for different lattice gases, different spatial dimensions and different relations between the time and the characteristic diffusion time , where . The simplest case turns out to be and . In this limit the leading-order results for the survival probability come from the steady-state solution of the MFT equations which has zero flux MVK (). Being interested in arbitrary , we will assume here that the leading-order results come from a family of stationary solutions of the MFT equations which are parameterized by the particle flux into the absorber. A different name for the stationarity assumption is additivity principle. This term was coined in Ref. Bodineau2004 () which studied the statistics of current in nonequilibrium steady states (NESS) in a finite one-dimensional setting.
This work mostly focuses on the SSEP. In the SSEP, each particle can hop, with an equal probability, to a neighboring lattice site if that site is unoccupied by another particle. If it is occupied, the hop is forbidden. At the coarse-grained level, the SSEP is described by Eq. (2) with and Spohn (); dimensions (). For an infinite SSEP with a spherical absorber, we expect the additivity principle to hold at arbitrary .
Before focusing on the SSEP we present, in Section II, the MFT formulation of the absorption statistics problem for an arbitrary diffusive gas at . Section III specifies the problem to the SSEP. A simple change of variables maps this problem into a universal problem of motion of an effective classical particle in a time-independent potential. This effective mechanical problem is solved in Section IV, where we evaluate and find the optimal density profile of the gas for arbitrary and . In the limit of non-interacting RWs, and the corresponding optimal density profile are determined in the Appendix.
Of special interest is the limit of where, as we show for the SSEP, . As expected, this probability density is much smaller than what is predicted by the Poisson distribution, observed for the RWs: .
We also show that, for , the optimal density profile of the SSEP, conditional on the absorption of particles, is monotonic in space at any . For the profile becomes non-monotonic when exceeds a critical value depending on , see Eq. (63).
Finally, we establish a close connection between the particle absorption statistics of the SSEP in the infinite space, considered here at , and the statistics of current in a finite SSEP in contact with two reservoirs at . We show that, when properly interpreted and rescaled, the moment generating functions of these two problems coincide.
We discuss our results and their possible extensions in Section VI.
Ii MFT of particle absorption: General
The MFT has become a standard framework for studying large deviations in diffusive lattice gases, see Ref. MFTreview () for a recent review. In the MFT, the particle number density field and the canonically conjugate “momentum” density field obey the Hamilton equations
is the Hamiltonian, and
is the Hamiltonian density. The spatial integration in Eq. (6), and everywhere in the following, is performed over the whole infinite space outside the absorber. Because of the spherical symmetry of the problem, we assume that and can only depend on the radial coordinate and time. The boundary conditions on the absorber are Bertini (); Tailleur (); MR ()
Far away from the absorber the gas is unperturbed:
where is the surface area of the -dimensional unit sphere, and is the gamma function.
At the level of individual realizations of the stochastic process, the gas density at can be either deterministic or random. In the former case (called the quenched case) one simply has
We will study the long-time particle absorption statistics in dimensions. In this case, the average particle flux to the absorber can be found by using the stationary solution of the diffusion equation (1). In the case of a spherical absorber, the stationary solution obeys the equation
Solving it with the boundary conditions and , one obtains in implicit form:
The long-time behavior of the average number of absorbed particles can now be found by multiplying the particle flux to the absorber by time. The result is
In particular, for (as it happens for the non-interacting RWs, for the SSEP and for the KMP model), Eq. (14) becomes the Laplace’s equation leading to
We argue that, at , fluctuations of the number of absorbed particles also come from a stationary solution, but this time it is the stationary solution of the MFT equations (3) and (4) which account for fluctuations. In other words, we assume that the additivity principle, postulated in Refs. Bodineau2004 (); DDR () in a finite system with two reservoirs, holds at for the infinite system with one absorber. The stationary solution yields, in the leading order of theory, the optimal density profile of the system, conditional on the number of absorbed particles . Once the steady state solutions and are found, we can calculate the action which yields up to a pre-exponential factor:
Notice that the steady-state solutions do not obey the boundary conditions in time, Eq. (11) or (12), and Eq. (13). To accommodate these conditions, the true time-dependent solution of the problem develops two narrow boundary layers in time, at and that give a subleading contribution to the action, cf. Ref. MVK ().
where we have set the negative arbitrary constant in Eq. (20) to , so that . The number of absorbed particles can be expressed via as follows:
Equation (20) yields
Plugging this into Eq. (21) we obtain
is the spherically symmetric Laplace operator in dimensions. There are two limits worth mentioning here:
Iii SSEP: Mechanical analogy
In its turn, the absorption probability density (27) reduces to
Fortunately, the nonlinear second-order equation (28) can be solved in elementary functions in any dimension. Let us define new variables and . The resulting equation for ,
is independent of and . It describes one-dimensional motion of an effective classical particle with unit mass ( is the “coordinate” of the effective particle, is “time”) in the potential
The energy integral is
The original boundary conditions and become and , respectively. That is, our effective particle must depart at from the point with the coordinate , where , and reach the origin at time .
Using Eq. (32), we obtain
where is rescaled energy of the effective particle, to be determined from the condition
The rescaled potential is equal to , see Fig. 1.
Equations (33) and (34) assume that the effective particle only moves to the left along the -axis, so that the resulting density profile is monotonic. This assumption is always correct for , that is . Here, in order to reach the origin, the effective particle must have a positive energy, , and move to the left, see Fig. 1.
For , that is , the gas density profile is only monotonic at sufficiently small , both for positive and negative . At sufficiently large it becomes non-monotonic and develops a local maximum which is higher than . Here the effective particle (with ) first moves to the right, is reflected from the potential barrier and then moves to the left and reaches . Here, instead of Eq. (34), we need to determine from the equation
where obeys the relation , that is . Correspondingly, , and subsequently , should be found from the following two equations:
describing the effective particle moving to the right and to the left, respectively. Here
The smaller is , the more pronounced the non-monotonicity of becomes at large . It is not surprising, therefore, that the non-monotonicity is also present in the model of non-interacting RWs, see the Appendix.
In the variables and , the probability distribution (30) becomes
The rescaled large deviation function is independent of , and . For concreteness, we will assume when presenting the formulas in dimensional (non-rescaled) form.
Iv SSEP: Solution
Before presenting the complete solution, let us consider three special cases.
iv.0.1 Mean-field limit
iv.0.2 Survival limit
hence . Plugging this value into Eq. (33) and again neglecting the , we obtain . This leads to the optimal gas density profile for survival:
and to the survival probability
in agreement with Ref. MVK ().
For (that is, for ) the limit of , or , corresponds to . Here the effective particle (see Fig. 1) moves to the left (if ), reaches the point (that is, ) and spends a very long time there before finally passing through and reaching the origin. The resulting stays close to on most of the interval and has two narrow boundary layers at and . The part of the trajectory where dominates the contribution to the probability density (37), and we obtain
independently of . A dependence on appears when we return from to , because of the relation .
Similarly, for (that is, for ), the effective particle with energy moves to the right, reaches the reflection point which is very close to , spends a very long time there and then gets reflected, moves to the left and reaches the origin. Again, the leading contribution to is described by Eq. (41), independently of . Back in the physical variables we see that, when the gas needs to pass a very large flux to the absorber, its density stays close to the half-filling value where is maximal, thus maximizing the fluctuation strength. The boundary layer at becomes a boundary layer at , whereas the boundary layer at spreads out to an infinite region .
Now we determine the full absorption statistics. We first consider the case of .
Here and . The effective particle can only move to the left: no reflection is possible. Evaluating the integral in Eq. (34), we obtain
and . As varies from zero to infinity, the function monotonically decreases from plus infinity to zero. Therefore, for the density profiles can be parameterized by . Note that as expected for the mean-field solution. The dependence of on is shown in Fig. 2.
The probability density (37) can be evaluated without calculating the optimal trajectory [or, in the original variables, the optimal density ]. Indeed, changing the integration variable in (37) from to and using the energy integral (32) and Eq. (42), we obtain
To remind the reader, here and . Evaluating the integral in Eq. (45), we obtain
This expression is valid for all , once we allow complex-valued functions at intermediate stages of evaluation. Equations (42) and (46) determine the probability distribution in a parametric form. Its asymptotics are
As expected, the distribution is peaked at the mean-field value . The distribution variance, as represented by the Gaussian asymptotic in the first line, reaches its maximum at . The leading first term at corresponds to the survival probability MVK (). The asymptotic at comes from the region where , that is , see Eq. (41).
whereas the asymptotics (47) become
Figure 3 depicts the probability density for , or .
What is the optimal stationary density profile for given and , that is for given and ? Evaluating the integral in Eq. (33), and going back to the original variables, we obtain
Another simple case is . Here we obtain
is again determined by Eq. (55), but now .
Here . At , where the critical value will be found shortly, the effective particle only moves to the left, and the resulting density profile is monotonic. In this regime the effective energy can take any value between and . Evaluating the integral in Eq. (34), we obtain
This expression is valid for . For the expression gives a complex number, and one should take its real part. The critical value is achieved at :
As expected, grows with and diverges at . Indeed, at no reflections are possible, and the optimal density profile is monotonic for any . Correspondingly, by sending to in Eqs. (57) and (58), one recovers Eq. (56) for .
For and the density profile is non-monotonic. Here , and Eq. (35) yields
A graph of is shown in Fig. 5.
Now we transform from to in the integral Eq. (37) and account for the two parts of the trajectory: before and after the reflection. The result is
Equations (57), (58), (60) and (62) determine the probability distribution versus for and arbitrary . To go over from to , as in Eqs. (48) and (49), one should use the relation [see Eq. (22) for ]. The critical value determines the critical number of absorbed particles
V Universality of the absorption statistics
In most of the paper we have dealt with the absorption probability distribution , and the rescaled large deviation function , see Eq. (37). An alternative description of the absorption statistics is in terms of a rescaled moment generating function , the Taylor expansion of which at yields the distribution cumulants, see e.g. Ref. Bodineau2004 (). A natural definition of , for , is the following:
where the averaging is with the distribution . A saddle-point calculation yields
where is the rescaled action defined in Eq. (37). The final result for is
Let us compare this result with the rescaled moment generating function that describes the statistics of current in a one-dimensional SSEP: a chain of lattice sites, connected at its two ends to two point-like reservoirs at densities and . The generating function is defined as follows:
As we can see, and coincide exactly if we identify the spherical absorber with one reservoir [and set in Eq. (70)] and identify infinity with the other reservoir [and set ]. This coincidence is unexpected because the two settings, the finite and infinite, are different. Moreover, was obtained for , whereas our does not apply for , where the optimal density profile is time-dependent MVK (). The coincidence of and is even more interesting in view of the fact that the generating function also describes the full counting statistics of free fermions transmitted through multichannel disordered conductors Beenakker (); Blanter (); Levitov ().
The formal reason why becomes clear in the mechanical analogy of Sec. III. Indeed, repeating our derivation for the finite one-dimensional setting we again arrive at Eq. (31), except that now , whereas the flux is replaced by the (minus) rescaled current . For we obtain
which coincides with Eq. (37) up to rescaling. In particular, the optimal density profiles in the two settings coincide up to the coordinate transformation .
Assuming the additivity principle, we evaluated the long-time probability distribution of absorption of the SSEP by a spherical absorber in an infinite space.
In the low-density limit, the exclusion effects can be neglected (see the Appendix) and, for not too large , can be approximately described by the Poisson distribution with mean . For finite , is strongly non-Poissonian. In particular,