The flashing Brownian ratchet and Parrondo’s paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.


Introduction
The flashing Brownian ratchet was introduced by Ajdari & Prost [1]; see also Magnasco [2]. It is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion, as explained in figure 1 (from Harmer et al. [3]) and figure 2 (from Parrondo & Dinís [4]). Earlier versions of these figures appeared in Rousselet et al. [5] and Faucheux et al. [6]. For another version, see Amengual [7, fig. 2.3].
The flashing Brownian ratchet is of interest not just to physicists but also to biologists in connection with so-called molecular motors (e.g. Bressloff [8, ch. 4]). The flashing Brownian ratchet is the process that motivated Parrondo's paradox [9,10], in which two fair games of chance, when alternated, produce a winning game.
Our aim here is to show, via a precise mathematical formulation of the flashing Brownian ratchet, how one can study the process numerically using a random walk approximation.  figure), the potential is on and all the particles are located around one of the minima of the potential. Then the potential is switched off and the particles diffuse freely, as shown in the centred figure, which is a snapshot of the system immediately before the potential is switched on again. Once the potential is connected again, the particles in the darker region move to the right-hand minimum whereas those within the small gray region move to the left. Due to the asymmetry of the potential, the ensemble of particles move, on average, to the right.' (Reprinted from Parrondo & Dinís [4, p. 148]  Using the notation of figure 1, it is clear how to formulate the model. First, the asymmetric sawtooth potential V is given by the formula (1.1) extended periodically (with period L) to all of R. Here 0 < α < 1 and L > 0, and asymmetry requires only that α = 1 2 . (α is a shape parameter and L is a scale factor; the latter is not important and some authors take L = 1.) The Brownian ratchet is a one-dimensional diffusion process with diffusion coefficient 1 and drift coefficient μ proportional to −V , that is, for some γ > 0, again extended periodically (with period L) to all of R. Such a process X t is governed by the stochastic differential equation (SDE): dX t = dB t + μ(X t ) dt, (1.2) where B t is a standard Brownian motion. This diffusion process drifts to the left on [nL, (n + α)L) and drifts to the right on [(n + α)L, (n + 1)L), for each n ∈ Z. In other words, it drifts towards a minimum of the sawtooth potential V. Given τ 1 , τ 2 > 0, the flashing Brownian ratchet is a time-inhomogeneous one-dimensional diffusion process that evolves as a Brownian motion on [0, τ 1 ] (potential 'off'), then as a Brownian ratchet on [τ 1 , τ 1 + τ 2 ] (potential 'on'), then as a Brownian motion on [τ 1 + τ 2 , 2τ 1 + τ 2 ] (potential 'off'), then as a Brownian ratchet on [2τ 1 + τ 2 , 2τ 1 + 2τ 2 ] (potential 'on') and so on. Such a process Y t is governed by the SDE: Note that, once the parameters of the sawtooth potential (α and L) are specified, the flashing Brownian ratchet is specified by three parameters, γ , τ 1 and τ 2 . (Alternatively, we could let the diffusion coefficients of the Brownian motion and the Brownian ratchet be σ 2 instead of 1, and then take τ 1 = 1 and τ 2 > 0.) Our formulation is equivalent to that of Dinís [14, eqn (1.78)], though parametrized differently. Occasionally, we may want to wrap these processes (the Brownian ratchet and the flashing Brownian ratchet) around the circle of circumference L. Because they are spatially periodic with period L, the wrapped processes remain Markovian. For example, we could define the wrapped Brownian ratchet X t byX t := e (2πi/L)X t .
Instead, we simply define it as the [0, L)-valued process with the understanding that the endpoints of the interval [0, L) are identified, effectively making it a circle of circumference L. The same procedure applies to the flashing Brownian ratchet, yieldinḡ Y t := mod(Y t , L).

Parrondo games from Brownian ratchets
We first consider the periodic drift coefficient μ described above in the case in which α = 1 3 and L = 3. We want to discretize space and time. We replace each interval [j, j + 1) by its midpoint j + 1 2 , which we relabel as j. In terms of μ, we define the discrete drift by μ j := μ(j + 1 2 ). Note that μ j = μ 0 < 0 if mod(j, 3) = 0 and μ j = μ 1 > 0 if mod(j, 3) = 1 or 2 ( figure 3). This discretizes space, now interpreted as profit in a game of chance instead of displacement. When the potential is off, we replace the Brownian motion by a simple   6]. Each interval [j, j + 1) (in black) is replaced by its midpoint j + 1 2 , which we relabel as j (in red) to discretize space. To discretize time as well, we replace the Brownian motion by a simple symmetric random walk on Z, and we replace the Brownian ratchet by an asymmetric random walk on Z whose periodic state-dependent transition probabilities are determined by a discretized version of μ. symmetric random walk on Z and call this game A, a fair coin-tossing game. When the potential is on, we replace the Brownian ratchet by an asymmetric random walk on Z whose periodic state-dependent transition probabilities are determined by the discrete drift and call this game B.
It should be mentioned that Pyke [16] found an elegant way to derive Parrondo's games (2.1) from a one-dimensional diffusion process that can be interpreted as a Brownian ratchet but with the sawtooth potential having a shape different from (1.1).
Thus, game A and (the generalized) game B lead to a more general form of Parrondo's paradox. In the conventional formulation, α is the reciprocal of an integer.

Approximating the Brownian ratchet
As in §2, let 0 < α < 1 and assume that α is rational, so that there exist relatively prime positive integers l and L with α = l/L. Consider a sequence of asymmetric random walks on Z with periodic state-dependent transition probabilities as follows. Given n ≥ 1, we let and P n (j, j − 1) = 1 − P n (j, j + 1), where as in (2.7). Note that the special case of (3.1) in which n = 1 is precisely (2.4). We want to let n → ∞ but first we let where λ > 0, then we rescale time by allowing n 2 jumps per unit of time, and finally we rescale space to {i/n : i ∈ Z} by dividing by n. The result in the limit as n → ∞ is a Brownian ratchet. Let D R [0, ∞) denote the space of real-valued functions on [0, ∞) that are right-continuous with left limits, and give it the Skorokhod topology.
Proof. The generator of the diffusion process satisfying the SDE (1.2) is and μ is extended periodically (with period L) to all of R. Then, by virtue of the Girsanov transformation, the martingale problem for L is well posed (e.g. [19,Theorem 6.4.3]) and it suffices to show that the discrete generator L n , given by Here we are using a result of Ethier & Kurtz [20,Corollary 4.8.17]. If {μ n } is a sequence of real numbers converging to μ, then we find that μ = −λ(1 − α)/(2α), and with we find that μ = λ/2. This suffices to complete the proof. (  We assume now that the time parameters τ 1 > 0 and τ 2 > 0 of the flashing Brownian ratchet are rational. Let m be the smallest positive integer such that m 2 τ 1 and m 2 τ 2 are integers. . .} denote the time-inhomogeneous random walk on Z that evolves as the simple symmetric random walk for n 2 τ 1 steps, then as the random walk of theorem 3.1 for n 2 τ 2 steps, then as the simple symmetric random walk for n 2 τ 1 steps, then as the random walk of theorem 3.1 for n 2 τ 2 steps, and so on. Let Y t denote the flashing Brownian ratchet with parameters γ = λ(1 − α)/2, The assumption about m ensures that the times n 2 τ 1 and n 2 τ 2 are integers. By Donsker's theorem applied to the simple symmetric random walk, Alternating in this way leads to the stated conclusion.  Table 1. Computations for the nth random walk (n = 100) approximating the flashing Brownian ratchet with α = 1 4 , L = 4, γ = 3λ/8 for various λ, τ 1 = τ 2 = 2.4, and initial state 0, at time τ 1 + τ 2 , illustrating the effect of varying the strength γ of the drift of the Brownian ratchet. To model figure 1 accurately, some measurements are needed. We begin with a cropped .pdf version of the figure and enlarge it on the computer screen to 800% of normal. It appears that the figure is rasterized, so our precision is limited. We measure that L = 206 mm and αL = 52 mm. Thus, we imagine that α = 1 4 was intended, and either the drawing or the measurements of it are slightly in error. We also measure the height of the normal curve at three places, namely 0, 1 and −3, assuming α = 1 4 and L = 4. We measure the respective heights to be 99.5 mm, 81 mm and 15 mm. Theoretically, the three heights are (2π t) −1/2 , (2π t) −1/2 e −1/(2t) , and (2π t) −1/2 e −9/(2t) . Therefore, we need to find t such that 99.5 e −1/(2t) = 81 and 99.5 e −9/(2t) = 15.
The equations have solutions t = 2.43062 and t = 2.37830, respectively. Because of the crudeness of our measurements, we round off to t = 2.4.
We conclude that the flashing Brownian ratchet described in figure 1 evolves as a Brownian motion (starting at 0) for time τ 1 = 2.4. Then the Brownian ratchet with α = 1 4 , L = 4 and γ to be specified runs (starting from where the Brownian motion ended) for time τ 2 to be specified. There is no good way to estimate γ and τ 2 from figure 1. We take τ 2 = τ 1 = 2.4 for convenience and let γ = λ(1 − α)/2 = 3λ/8 for several choices of λ (λ = 1, 2, 3, 4, 5). Then the Brownian motion runs (starting from where the Brownian ratchet ended) for time τ 1 = 2.4, then the Brownian ratchet runs for time τ 2 = 2.4, and so on. We are interested in the distribution of the process at time τ 1 + τ 2 = 4.8, which we can compare with the third panel in figure 1.
There are several notable differences between the figures of figure 4 and the third panel of figure 1. First, the three peaks of the density are pointed, unlike a normal density, so figure 2 is more accurate in this regard. Second, they are asymmetric, with more mass to the left than to the right of −4, 0 and 4. Presumably, the explanation for this is that, for example, the drift to the left on [0, 1) is stronger than the drift to the right on [−3, 0). Another distinction is that the ratio of the height of the highest peak to that  Table 2. Computations for the nth random walk (for various n) approximating the flashing Brownian ratchet with α = 1 4 , L = 4, γ = 3λ/8, λ = 5, τ 1 = τ 2 = 2.4, and initial state 0, at time τ 1 + τ 2 , illustrating the rate of convergence in a special case of theorem 3.  Consider the case λ = 5. The areas under the three peaks of the density are, respectively, 0.0330104, 0.731102 and 0.235888. (These numbers are exact, not for the flashing Brownian ratchet, but for our random walk approximation to it, with n = 100.) If the peaks were symmetric, the mean displacement would be (−4)(0.0330104) + (0)(0.731102) + (4)(0.235888) = 0.811510, but in fact the mean displacement is 0.678364 (again, an approximation) because of the asymmetry of each peak. Table 1 shows the effect of varying λ on several statistics of interest. We might ask whether, as suggested in figures 1 and 2, the areas of the three peaks are equal to the corresponding areas under the normal curve. The latter areas are where σ = √ 2.4. It seems evident that the answer is affirmative in the limit as λ → ∞ (table 1). We return to the case λ = 5. To get a sense of the rate of convergence in theorem 3.2, we provide in table 2  The density of π is a periodic function (with period L) whose maxima occur at the minima of the sawtooth potential.
Theorem 5.1. The Brownian ratchet with parameters α, L and γ has a reversible invariant measure π of the form (5.1). The wrapped Brownian ratchet with the same parameters has a reversible invariant measure of the same form, restricted to [0, L).
Proof. We use a different characterization of the Brownian ratchet. We take D(L ), the domain of L , to be the space of real-valued C 1 functions f on R with limits at ±∞ such that f is absolutely continuous and has a right derivative, denoted by f , with L f continuous on R with limits at ±∞. In particular, the discontinuities of f must be compatible with those of the drift coefficient μ. Thus, for all n ∈ Z. Mandl for every f , g ∈ D(L ) with compact support. The third equality uses integration by parts, the fourth uses V(nL) = 0 for all n ∈ Z and the fifth uses a telescoping sum and the compact support assumption. The right side of (5.2) is symmetric in f and g, so the left side must be too, and we have for every f , g ∈ D(L ) with compact support, as required.
For the second assertion, we take D(L ) to be the space of real-valued C 1 functions f on the circle [0, L) such that f is absolutely continuous and has a right derivative, denoted by f , with L f continuous on the circle [0, L). Thus, Finally, for every f , g ∈ D(L ), as in (5.2) except with R replaced by For both reversible invariant measures (unrestricted and restricted), we expect there is a uniqueness result but we currently lack a proof. Note that the mean drift, with respect to the reversible invariant probability measure, is equal to Thus, the mean drift is 0 at equilibrium (of the wrapped Brownian ratchet).
Denote the flashing Brownian ratchet at time t, starting from x ∈ R at time 0, by Y x t , and the wrapped flashing Brownian ratchet at time t, starting from x ∈ [0, L) at time 0, byȲ x t . Then the one-step transition functionP for a continuous-state Markov chain has a stationary distributionπ . (Existence is automatic from the Feller property and the compactness of the state space; recall that the endpoints of [0, L) are identified. Nevertheless, no analytical formula is known, and uniqueness is expected but unproved.) The mean displacementμ of the flashing Brownian ratchet over the time interval [0, τ 1 + τ 2 ], starting from the stationary distributionπ , namelyμ is a statistic of primary interest. The second equality is a consequence of the periodicity of the integrand (with period L) and the convention that we do not distinguish notationally betweenπ and its image under the mapping The advantage of modifyingπ in this way is that, when regarded as a measure on R, it becomes unimodal instead of U-shaped. We propose to approximateμ as follows. The integrand can be estimated as before, the only difference being that the starting point of the flashing Brownian ratchet is x, not 0. The stationary distributionπ of the one-step transition function (5.3) can be approximated by the stationary distribution of the finite Markov chain whose one-step transition matrix has the form P(i, j) := P(Ȳ n (n 2 (τ 1 + τ 2 )) = j |Ȳ n (0) = i), i, j = 0, 1, . . . , nL − 1, where {Ȳ n (k), k = 0, 1, . . .} denotes the wrapped (period nL) random walk used to approximate the wrapped flashing Brownian ratchet. A small technical issue, if nL is even, is that this Markov chain fails to be irreducible if n 2 (τ 1 + τ 2 ) is even, in which case we replace the latter quantity by n 2 (τ 1 + τ 2 ) + 1. Then the chain becomes irreducible and there is a unique stationary distribution. The black curve of the first panel of figure 5 is an approximation to the density ofπ with support [−3, 1) instead of [0, 4). Starting from the approximateπ at time 0, the second and third panels show the approximations to the density at times τ 1 and τ 1 + τ 2 , respectively. Figure 5   The blue curves represent the sawtooth potential. The vertical axes in the first and third panels are comparable, whereas the vertical axis in the second panel has been stretched for clarity.

Conclusion and future work
A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a time-inhomogeneous one-dimensional diffusion process that alternates between a Brownian motion and a Brownian ratchet. We propose a random walk approximation to the Brownian ratchet and the flashing Brownian ratchet. This provides an efficient method of numerically studying these continuous processes, and furthermore it is more accurate than a simulation, based on a random number generator, would be. By using the random walk approximation, we find the approximate density of the flashing Brownian ratchet after one time period, starting at 0. We also find the approximate density of the flashing Brownian ratchet after the same time period, but now starting from a stationary distribution associated with the so-called wrapped flashing Brownian ratchet, and we approximate the mean displacement of the flashing Brownian ratchet over that time period. The goal was to determine how accurate the conceptual figures 1 and 2 are. We began by deriving a general class of capital-dependent Parrondo games motivated by the Brownian ratchet with shape parameter α. It has been conventional to assume that α is the reciprocal of an integer, but we allow it to be an arbitrary rational number in (0, 1). These Parrondo games, in turn, motivated our random walk approximation.
As for future work, we are currently trying to apply these ideas to what might be called a tilted flashing Brownian ratchet, that is, a flashing Brownian ratchet in the presence of a macroscopic gradient that reduces the directed motion effect. See fig. 6, (d)-(f), of Harmer & Abbott [9].