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Biased tracer diffusion in a high-density lattice gas
General method
We present here the general model of a biased tracer in a hardcore lattice gas. We consider a hypercubic lattice of arbitrary dimension. The lattice sites are occupied by hardcore particles performing symmetric random walks with a mean waiting time , with the restriction that the occupancy number for each site is at most equal to one. The fraction of occupied sites is denoted as , whereas the vacancy density is denoted as 0 (with 0 = 1 ). A tracer particle (TP) is also present on the lattice, and performs a nearest-neighbor biased random walk with a mean waiting time (in what follows, we will consider the particular case where = ). The probability for the TP to make one step in direction e will be denoted by p ( 2 f 1; : : : ; dg). Although our results will be valid for any choice of the probabilities p , it will be convenient to assume that the bias originates from an external force F = F e1, where the lattice spacing and the inverse temperature are taken equal to one. Note that this choice of the jump probabilities fulfills the detailed balance condition.
In the high-density limit, the motion of the TP is mediated by the diffusion of the vacancies: the TP can move only if a vacancy visits one of its neighboring sites. In this situation, it is easier to describe the dynamics of the vacancies rather than the dynamics of the whole bath of particles. We adopt a discrete-time description, and at each time step, each vacancy moves according to the following rules: (i) if the TP is not adjacent to the vacancy, one neighbor is randomly selected and exchanges its position with the vacancy; (ii) otherwise, if the TP is at position X and the vacancy at position X + e , the TP exchanges its position with the vacancy with probability p = =[1 1=(2d) + p ] and with probability 1=(2d 1 + 2dp ) with any of the 2d 1 other neighbors. We aim to obtain results at leading order in the density of vacancies 0. Consequently, as the events involving two vacancies at the same site or on neighboring sites are of order O( 02), they do not contribute in the limit we consider and are not taken into account.
In the spirit of [22, 23], where tracer diffusion in the absence of bias was studied, we first consider an auxiliary problem involving a single vacancy, starting from site Y 0. The TP, initially at site 0, can move only by exchanging its position with the vacancy. For the sake of simplicity, we first present in detail the particular case where the applied force is strong enough for the TP motion to be directed, so that p1 = 1 and p = 0 for 6= 1. Results in the general case of an arbitrary force, obtained along the same method, will be given next. We define the single-vacancy propagator Pt(1)(XjY 0) as the probability for the TP to be at site X = ne1 at time t knowing that the vacancy started from site Y 0. An expression for this quantity can be obtained by summing over the number of steps taken by the TP up to time t, over the directions of these steps, and over the length of the time intervals elapsed between consecutive steps.
Consequently, the determination of the function (k; ) and therefore of the cumulants of the position of the TP only relies on the FPTD Ft(Y 0) (probability for a vacancy to visit the origin for the first-time at time t starting from site Y 0).
When the tracer is no longer directed and undergoes an arbitrary bias, we show that the single vacancy propagators and the ultimate expression of (k; ) only depends on conditional FPTD Ft (0je jY 0) (probability for a vacancy to visit the origin for the first-time at time t starting from Y 0 and being at site e at time t 1). These conditional FPTD are relative to the random walk of a vacancy on a lattice in presence of a biased TP that locally modifies the evolution rules of the vacancy. The va-cancy then performs a symmetric random walk on every sites of the lattice, excepted on the neighboring sites of the TP, on which the evolution rules of the vacancies are perturbed because of the bias experi-enced by the TP. The conditional FPTD are computed using standard techniques for random walks on lattices with defectives site [58], and written in terms of the propagators associated to the simple sym-metric random walk on the considered structure. These calculations yield explicit expressions for the conditional FPTD but are very lengthy, so that the help from a computer algebra software is required.
In this framework, we will study the fluctuations of the position of the TP as well as the other cumulants of its position. The dependence of these quantities on the different parameters of the problem (time, force, density, lattice dimension) are shown to be non-trivial. We first consider the case of a one-dimensional lattice, for which we determine the probability distribution of the TP position. We consider higher-dimensional lattices, and study the fluctuations and mean of the position of the TP. We then study the higher-order cumulants of the distribution, and show that there exists universal formulae to describe their behavior in terms of the properties of a simple random walk on the considered structures. Finally, we provide a simplified description that correctly captures the physical mechanisms at stake.
One-dimensional lattice
We first consider the case of a one-dimensional lattice. The transport properties of a biased TP in a one-dimensional hardcore lattice gas are related to the well-known problem of single-file diffusion. In this paradigmatic model, all the particles are identical and perform symmetric nearest-neighbor random walks with the hardcore exclusion constraint. In the absence of a bias, it has beenpshown that the fluctuations of a tagged particle are anomalous and that they grow subdiffusively as t [54, 4]. The situation where the tagged particle is biased also raised some attention. In this case, the mean position p itself has a nontrivial behavior, and it was shown that it was growing as t with a prefactor that is a function of the bias and which was given as the implicit solution of an equation [25, 72].
In the presence of a bias, the fluctuations of the position of the TP and its distribution have not been studied yet. In the high-density limit, we use the general framework presented in the previous section. We express the cumulant generating function of the TP position in terms of the first-passage time densities (FPTD) associated to the random walk of the vacancies. These FPTD are calculated with standard methods from the theory of random walks on lattices. It is shown that all the odd cumulants on the one hand, and all the even cumulants on the other hand are identical in the long-time limit,
Confinement-induced superdiffusion
We now consider higher-dimensional lattices. As it was emphasized in Section 0.2.1, an important technical point is that for small values of the density of vacancies 0, the dynamics of the TP can be
deduced from analyzing the joint dynamics of the TP and a single isolated vacancy. Exact asymptotic expressions of the fluctuations of the TP position in the direction of the force, denoted by (2)1(t), are obtained for various geometries and for arbitrary values of the jump probabilities p (probability for the TP to make a jump in direction ). These are valid at large times and low vacancy densities, and are summarized below.
Strong superdiffusion with exponent 3=2 takes place in confined, quasi-1D geometries, those being, infinitely long 3D capillaries and 2D stripes. This result is quite counterintuitive: indeed, in the absence of driving force it is common to encounter diffusive, or even subdiffusive growth of the fluctuations of the TP position in such crowded environments, however not superdiffusion.
The superdiffusion in such systems emerges beyond (and therefore can not be reproduced within) the linear response-based approaches. If the bias originates from an external force F and if the jump probabilities of the TP are given by (1), the prefactor in the superdiffusive law is proportional to F 2 when F ! 0. Despite the presence of the superdiffusion, the Einstein relation is nonetheless valid for systems of arbitrary geometry due to subdominant (in time) terms whose prefactor is proportional to F .
In unbounded 3D systems (2)1(t) grows diffusively and not superdiffusively.
Finally, this shows that superdiffusion is geometry-induced and the recurrence of the random walk performed by a vacancy is a necessary condition in order for superdiffusion to occur. However, this condition is not sufficient. Indeed, on one-dimensional lattices, although the random walk of a vacancy is recurrent, the behavior of the TP is not superdiffusive (see Section 0.2.2).
Giant diffusion regime. The exact analytical result in (10) provides explicit criteria for superdiffusion to occur. Technically, this yields the behavior of the variance when the limit 0 ! 0 is taken before the large-t limit. It however does not allow us, due to the nature of the limits involved, to answer the question whether the superdiffusion is the ultimate regime (or just a transient), which requires the determination of limt!1 (2)1(t) at fixed 0. Importantly, we find that the order in which these limits are taken is crucial in confined geometries and show that limt!1 lim 0!0 (2)1(t) 6= lim 0!0 limt!1 (2)1(t). In fact, the effective bias experienced by a vacancy between two consecutive interactions with the TP, originating from a non zero velocity of the TP, dramatically affects the ultimate long-time behavior of the variance in confined geometries.
More precisely, we show that the superdiffusive regime is always transient for an experimentally relevant system with 0 fixed, while the long-time behavior obeys On quasi-one-dimensional and two-dimensional lattices, the crossover times between the two regimes scales as 1= 02, so that superdiffusion is very long-lived in these systems. On Fig. 3, we present results from Monte-Carlo numerical simulations performed on a two-dimensional stripe of width L = 3, in the case where the TP is directed. The fluctuations of the position of the TP divided by time are plotted as a function of a rescaled time = a002 02t (where a00 is a function of the bias which will be introduced explicitly in the next section) for different values of the density of vacancy 0. For small values of the rescaled time ( 1), the fluctuations grow superdiffusively as t3=2. At long times ( 1), fluctuations cross over to a diffusive regime. The scaling function fe( )=L is also plotted (black line).
We verify that it constitutes a good description of both regimes and of the transition between them.
The superdiffusive behavior of the biased TP was also observed in off-lattice systems. The behavior of a driven TP in an continuous-space bath of Brownian particles was studied by numerical simulations in two types of system: (i) a colloidal fluid of identical particles interacting via a purely repulsive potential (simulations performed by A. Law and D. Chakraborty, Universität Stuttgart), (ii) a dissipative granular fluid (simulations performed by A. Bodrova, Moscow State University). In both algorithms, a biased intruder is submitted to an external force. In stripe-like geometries, the simulations reveal a superdiffusive behavior of the position of the TP, with fluctuations growing as t3=2. This suggests that confinement-induced superdiffusion could be a generic feature of the dynamics of a biased intruder in a crowded medium.
Velocity anomaly in quasi-one-dimensional geometries
We found that in quasi-one-dimensional and two-dimensional systems there exists a long-lived superdif-fusive behavior of the fluctuations of the position of the TP, crossing-over to a diffusive behavior after a time t 1= 02. The complete time behavior of the variance was found to display a scaling behavior as a function of the rescaled variable 02t. We show that, actually, the behavior of the mean itself of the position of the TP displays a striking anomaly in quasi-one-dimensional geometries. This unexpected behavior, obtained from Monte-Carlo numerical simulations in a quasi-1D stripe, is plotted in Fig. 4 for several vacancy densities 0, as a function of the rescaled variable = a002 02t, suggested by the scaling behavior of the variance. A scaled form of the mean-position is found which, very surprisingly, after a long-lived plateau drops to a lower ultimate value. The transition from the “high” velocity to the ultimate regime of “low” velocity takes place at a time scale of the order of the cross-over time t 1= 02 involved in the time evolution of the variance, suggesting that the anomaly of the variance and that of the mean could be linked.
Using again the general formalism valid at leading order in 0 presented in Section 0.2.1, we study analytically the mean position of the TP. One can obtain the expression of the mean position of the TP in terms of the conditional FPTD associated to the random walks of the vacancies, which have been computed for the study of the fluctuations of the TP.
Table of contents :
0 Main results
0.1 Introduction
0.2 Biased tracer diffusion in a high-density lattice gas
0.2.1 General method
0.2.2 One-dimensional lattice
0.2.3 Confinement-induced superdiffusion
0.2.4 Velocity anomaly in quasi-one-dimensional geometries
0.2.5 Universal formula for the cumulants
0.2.6 Simplified description
0.3 Biased tracer diffusion in a hardcore lattice gas of arbitrary density
0.3.1 General formalism
0.3.2 One-dimensional situation
0.3.3 Higher-dimensional lattices
0.4 Conclusion
1 Introduction
I Biased tracer diffusion in a high-density lattice gas
2 Motivation and general presentation
2.1 Introduction
2.1.1 Statement of the problem
2.1.2 Experimental works
2.1.3 Theoretical descriptions
2.1.4 Objectives
2.2 Presentation of the model
2.2.1 Model
2.2.2 Evolution rules
2.3 Single-vacancy situation
2.3.1 Single-vacancy propagator
2.3.2 Calculation of the conditional first-passage time densities F
2.4 Finite low vacancy density
2.4.1 Average over the initial positions of the vacancies
2.4.2 Computation of the quantities F0
2.4.3 Thermodynamic limit and expression of the cumulants
2.5 Conclusion
3 One-dimensional geometry
3.1 Introduction
3.1.1 Single-file diffusion
3.1.2 Case of a biased TP
3.1.3 Objectives of this chapter
3.2 Resolution
3.2.1 Model
3.2.2 Cumulant generating function
3.2.3 Calculation of b F1
3.2.4 Calculation of h()
3.2.5 Expression of the cumulants
3.3 Results
3.3.1 Cumulants in the long-time limit
3.3.2 Numerical simulations
3.3.3 Case of a symmetric TP
3.3.4 Full distribution
3.4 Conclusion
4 Confinement-induced superdiffusion
4.1 Introduction
4.1.1 Context
4.1.2 Objectives of this Chapter
4.2 Main results of this Chapter
4.3 Stripe-like geometry
4.3.1 Expression of b (k1; )
4.3.2 Calculation of the conditional FPTD F
4.3.3 Calculation of the quantities F0
4.3.4 Expression of the second cumulant in the long-time limit
4.3.5 Comments
4.3.6 Numerical simulations
4.4 Capillary-like geometry
4.4.1 Introduction
4.4.2 Expression of b (k1; )
4.4.3 Conditional FPTD b F and sums F0
4.4.4 Expression of the second cumulant in the long time limit
4.4.5 Numerical simulations
4.5 Two-dimensional infinite lattice
4.5.1 Computation of the second cumulant
4.5.2 Long-time expansion
4.5.3 Subdominant term
4.5.4 Remarks and numerical simulations
4.6 Three-dimensional infinite lattice
4.7 Crossover to diffusion – Stripe-like geometry
4.7.1 Introduction
4.7.2 Determination of the conditional FPTD
4.7.3 Propagators
4.7.4 Ultimate expression of the second cumulant
4.7.5 Scaling function
4.8 Crossover to diffusion – Capillary-like geometry
4.8.1 Introduction
4.8.2 Ultimate expression of the second cumulant
4.8.3 Scaling function
4.9 Crossover to diffusion – Two-dimensional lattice
4.9.1 Ultimate expression of the second cumulant
4.9.2 Scaling function
4.10 Conclusion
5 Velocity anomaly in quasi-one-dimensional geometries
5.1 Introduction
5.2 Quasi-one-dimensional geometries
5.2.1 Stripe-like geometry
5.2.2 Capillary-like geometry
5.3 Two-dimensional lattice
5.4 Conclusion
6 Universal formulae for the cumulants
6.1 Introduction
6.2 First cumulants of the TP position
6.2.1 Mean position of the TP
6.2.2 Fluctuations of the position of the TP
6.3 Higher-order cumulants in the longitudinal direction
6.3.1 Method
6.3.2 Recurrent lattices
6.3.3 Transient lattices
6.4 Cumulants in the transverse direction
6.4.1 Method
6.4.2 Recurrent lattices
6.4.3 Transient lattices
6.5 Extension to fractal lattices
6.6 Conclusion
7 Simplified continuous description
7.1 Introduction
7.2 General formalism
7.3 Two-dimensional lattice
7.4 Stripe-like lattice
7.5 One-dimensional lattice
7.6 Conclusion
II Biased tracer diffusion in a lattice gas of arbitrary density
8 General formalism and decoupling approximation
8.1 Introduction
8.2 Model and master equation
8.2.1 Model
8.2.2 Master equation
8.3 Equations verified by the first cumulants
8.3.1 Mean position
8.3.2 Fluctuations of the TP position
8.3.3 Stationary values
8.4 Cumulant generating function
8.4.1 Governing equations
8.4.2 Evolution equations of the quantities ewr
8.4.3 Application: Third-order cumulant
8.5 Conclusion
9 One-dimensional lattice in contact with a reservoir
9.1 Introduction
9.2 First cumulants of the TP position in one dimension
9.2.1 Solution of the equation on kr in one dimension
9.2.2 Solution of the equation on egr in one dimension
9.2.3 Solution of the equation on emr in one dimension
9.3 Results and discussion
9.3.1 Algorithm and numerical methods
9.3.2 Velocity
9.3.3 Diffusion coefficient
9.3.4 Third cumulant
9.4 Cumulant generating function and propagator
9.4.1 Calculation
9.4.2 Numerical simulations
9.5 Conclusion
10 Resolution on higher-dimensional lattices
10.1 Introduction
10.2 Mean position of the TP
10.2.1 Basic equations
10.2.2 Infinite lattices
10.2.3 Generalized capillaries
10.2.4 General solution
10.2.5 High-density limit
10.2.6 Low-density limit and fixed obstacles
10.3 Negative differential mobility
10.3.1 Introduction
10.3.2 Simple physical mechanism
10.3.3 Method and results
10.3.4 Summary
10.4 Fluctuations of the TP
10.4.1 General equations
10.4.2 High-density limit
10.5 Conclusion
11 Conclusion
12 Publications
