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].

Figure 1.

‘Brownian ratchet mechanism. The sawtooth and flat potentials are labeled with U and U, respectively, while the distribution of Brownian particles is shown via the normal curves. This sequence of flashing between on and off potentials shows there is a net movement of particles to the right.’ (Reprinted from Harmer et al. [3, p. 706] with the permission of AIP Publishing.)

Figure 2.

‘The flashing ratchet at work. The figure represents three snapshots of the potential and the density of particles. Initially (upper 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] with the permission of Taylor & Francis Ltd.)

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. In §2, we provide a general formulation of Parrondo’s paradox motivated by the flashing Brownian ratchet. These Parrondo games are then modified in §3 so as to yield our random walk approximation. We determine, in §§4 and 5, whether the conceptual figures 1 and 2 accurately represent the behaviour of the flashing Brownian ratchet.

Alternatively, one could numerically solve a partial differential equation, specifically the Fokker–Planck equation, to obtain similar results, but we believe that our method is simpler. Discretization of the Fokker–Planck equation for the Brownian ratchet, and the relationship to Parrondo’s games, has been explored by Allison & Abbott [11] and Toral et al. [12,13].

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 extended periodically (with period L) to all of R. Here 0<α<1 and L>0, and asymmetry requires only that . (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 is governed by the stochastic differential equation (SDE): where B 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 τ1,τ2>0, the flashing Brownian ratchet is a time-inhomogeneous one-dimensional diffusion process that evolves as a Brownian motion on [0,τ1] (potential ‘off’), then as a Brownian ratchet on [τ1,τ1+τ2] (potential ‘on’), then as a Brownian motion on [τ1+τ2,2τ1+τ2] (potential ‘off’), then as a Brownian ratchet on [2τ1+τ2,2τ1+2τ2] (potential ‘on’) and so on. Such a process Y is governed by the SDE: where[1] Note that, once the parameters of the sawtooth potential (α and L) are specified, the flashing Brownian ratchet is specified by three parameters, γ, τ1 and τ2. (Alternatively, we could let the diffusion coefficients of the Brownian motion and the Brownian ratchet be σ2 instead of 1, and then take τ1=1 and τ2>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 by 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, yielding

Parrondo games from Brownian ratchets

We first consider the periodic drift coefficient μ described above in the case in which and L=3. We want to discretize space and time. We replace each interval [j,j+1) by its midpoint , which we relabel as j. In terms of μ, we define the discrete drift by . Note that μ=μ0<0 if mod(j,3)=0 and μ=μ1>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 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.

Figure 3.

The periodic drift μ with and L=3 is plotted on the interval [−6,6]. Each interval [j,j+1) (in black) is replaced by its midpoint , 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.

We find that the asymmetric random walk on Z has periodic state-dependent transition probabilities of the form and P(j,j−1)=1−P(j,j+1), where since μ=μ0<0 if mod(j,3)=0 and since μ=μ1>0 if mod(j,3)=1 or 2. Because of the periodic transition probabilities, the unique reversible invariant measure π must be periodic (i.e. π(j)=π(j+3)) for the random walk to be recurrent. We can check that the detailed balance conditions have a solution if and only if . Solving for p1 in terms of p0, we find that Denoting the square root in the denominator by ρ, the requirements that become 0<ρ<1, and (This is the parametrization of Ethier & Lee [15].) Further, in terms of ρ, the reversible invariant measure restricted to {0,1,2} has, via (2.2), the form resulting in a mean profit of π(0)(2p0−1)+(π(1)+π(2))(2p1−1)=0, so game B is also fair (asymptotically). Nevertheless, the random mixture cA+(1−c)B (0

There are several proofs available for these results, including Pyke [16], based on mod m random walks; Key et al. [17], based on random walks in periodic environments; Ethier & Lee [15], based on the strong-mixing central limit theorem; and Rémillard & Vaillancourt [18], based on Oseledec’s multiplicative ergodic theorem.

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).

The above formulation with and L=3 can be generalized. 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. Game A is as before, whereas game B is described by an asymmetric random walk on Z with periodic state-dependent transition probabilities of the form and P(j,j−1)=1−P(j,j+1), where as before. Because of the periodic transition probabilities, the unique reversible invariant measure π must be periodic (i.e. π(j)=π(j+L)) for the random walk to be recurrent. We can check that the detailed balance conditions have a solution if and only if . Solving for p1 in terms of p0, we find that Denoting the α/(1−α)th power in the denominator by ρ, the requirements that become 0<ρ<1, and Note that (2.4) and (2.7) generalize (2.1) and (2.3).

Further, in terms of ρ, the reversible invariant measure restricted to {0,1,…,L−1} has, via (2.5), the form where C is chosen so that π(0)+π(1)+⋯+π(L−1)=1, resulting in a mean profit of so game B is also fair (asymptotically).

As a function of p0 the function in (2.6) is strictly convex on if . It follows that the random mixture cA+(1−c)B (0

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(j,j−1)=1−P(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 but first we let where λ>0, then we rescale time by allowing n2 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 is a Brownian ratchet.

Let denote the space of real-valued functions on that are right-continuous with left limits, and give it the Skorokhod topology.

Theorem 3.1

For n=1,2,… (and n>λ), let {X(k), k=0,1,…} denote the random walk on defined by (3.1)–(3.3), and let X denote the Brownian ratchet with γ=λ(1−α)/2. If X(0)/n converges in distribution to X0 as , then {X(⌊n2t⌋)/n, t≥0} converges in distribution in to {X, t≥0} as .

Proof.

The generator of the diffusion process satisfying the SDE (1.2) is acting on , the space of real-valued functions on R with compact support, where and μ is extended periodically (with period L) to all of R. Then, by virtue of the Girsanov transformation, the martingale problem for is well posed (e.g. [19, Theorem 6.4.3]) and it suffices to show that the discrete generator , given by converges to in the sense that Here we are using a result of Ethier & Kurtz [20, Corollary 4.8.17].

If {μ} is a sequence of real numbers converging to μ, then uniformly over all x with nx∈Z, provided . With we find that μ=−λ(1−α)/(2α), and with we find that μ=λ/2. This suffices to complete the proof. (We leave it to the reader to check that the compact containment condition is satisfied.) ▪

We assume now that the time parameters τ1>0 and τ2>0 of the flashing Brownian ratchet are rational. Let m be the smallest positive integer such that m2τ1 and m2τ2 are integers.

Theorem 3.2

For n=m,2m,3m,… (and n>λ), let {Y (k), k=0,1,…} denote the time-inhomogeneous random walk on that evolves as the simple symmetric random walk for n2τ1 steps, then as the random walk of theorem 3.1 for n2τ2 steps, then as the simple symmetric random walk for n2τ1 steps, then as the random walk of theorem 3.1 for n2τ2 steps, and so on. Let Y denote the flashing Brownian ratchet with parameters γ=λ(1−α)/2, τ1>0 and τ2>0. If Y (0)/n converges in distribution to Y 0 as , then {Y (⌊n2t⌋)/n, t≥0} converges in distribution in to {Y , t≥0} as . (Here through multiples of m.)

The assumption about m ensures that the times n2τ1 and n2τ2 are integers. By Donsker’s theorem applied to the simple symmetric random walk, {Y (⌊n2t⌋)/n, 0≤t≤τ1} converges in distribution in D[0,τ1] to {Y , 0≤t≤τ1}. Then, by theorem 3.1, {Y (⌊n2t⌋)/n, τ1≤t≤τ1+τ2} converges in distribution in D[τ1,τ1+τ2] to {Y , τ1≤t≤τ1+τ2}. Alternating in this way leads to the stated conclusion. ▪

Density of the flashing Brownian ratchet at time τ1+τ2, starting at 0

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 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 and L=4. We measure the respective heights to be 99.5 mm, 81 mm and 15 mm. Theoretically, the three heights are (2πt)−1/2, (2πt)−1/2 e−1/(2, and (2πt)−1/2 e−9/(2. Therefore, we need to find t such that 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 τ1=2.4. Then the Brownian ratchet with , L=4 and γ to be specified runs (starting from where the Brownian motion ended) for time τ2 to be specified. There is no good way to estimate γ and τ2 from figure 1. We take τ2=τ1=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 τ1=2.4, then the Brownian ratchet runs for time τ2=2.4, and so on. We are interested in the distribution of the process at time τ1+τ2=4.8, which we can compare with the third panel in figure 1.

There is no known analytical formula for the density of the flashing Brownian ratchet at time τ1+τ2 (however, see Zadourian et al. [21]). But we can approximate it numerically as suggested in theorem 3.2. The positive integer m of that theorem is 5. In each case, we take n=100, meaning that at time τ1+τ2=4.8, the approximating random walk has made 4.8n2=48 000 steps. We compute its distribution recursively after 1 step, 2 steps, … , 48 000 steps, using the simple symmetric random walk for the first 24 000 steps, then the asymmetric random walk for the next 24 000 steps. Note that the distribution of the random walk after 2k steps is concentrated on {−2k,−2(k−1),…,0,…,2(k−1),2k}, whereas the distribution after 2k+1 steps is concentrated on {−2k−1,−2k+1,…,−1,1,…,2k−1,2k+1}. We save the distribution after 48 000 steps, plot the histogram, and interpolate linearly. The results are displayed in figure 4.

Figure 4.

We mathematically model the third panel of figure 1. A Brownian motion, starting at 0, runs for time τ1=2.4. Then, starting from where the Brownian motion ended, a Brownian ratchet with , L=4, γ=3λ/8 and λ=1,2,3,4,5 (from top to bottom) runs for time τ2=2.4. The black curve is an approximation to the density of the flashing Brownian ratchet at time τ1+τ2, via the random walk approximation with n=100. The blue curve is the sawtooth potential.

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 of the second highest is at least 3 in figure 4 (table 1) but is less than 1.5 in figure 1. While this is true for each λ=1,2,3,4,5, it may be partly a consequence of our arbitrary choice of τ2.

Table 1.

Computations for the nth random walk (n=100) approximating the flashing Brownian ratchet with , L=4, γ=3λ/8 for various λ, τ1=τ2=2.4, and initial state 0, at time τ1+τ2, illustrating the effect of varying the strength γ of the drift of the Brownian ratchet.

λ	areas of the three peaks	heights of the three peaks	mean displacement
1	(0.0688267,0.701114,0.230060)	(0.0627471,0.566531,0.121751)	0.0595931
2	(0.0500629,0.734941,0.214996)	(0.0756255,1.06860,0.274227)	0.297582
3	(0.0400379,0.737033,0.222929)	(0.0875995,1.59779,0.464698)	0.496585
4	(0.0354116,0.734036,0.230552)	(0.1021090,2.11341,0.657213)	0.611651
5	(0.0330104,0.731102,0.235888)	(0.117836,2.60974,0.839352)	0.678364
10	(0.0290537,0.723174,0.247772)	(0.197900,4.92657,1.68412)	0.809036
15	(0.0279536,0.719952,0.252094)	(0.273152,7.03601,2.45801)	0.853220
20	(0.0274363,0.718221,0.254343)	(0.342844,8.97601,3.17124)	0.875658
25	(0.0271326,0.717131,0.255736)	(0.407788,10.7794,3.83499)	0.889397
50	(0.0264993,0.714662,0.258839)	(0.695524,18.7599,6.77822)	0.919557

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 . It seems evident that the answer is affirmative in the limit as (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 computed values of several statistics as functions of n (n=10,20,30,…,200).

Table 2.

Computations for the nth random walk (for various n) approximating the flashing Brownian ratchet with , L=4, γ=3λ/8, λ=5, τ1=τ2=2.4, and initial state 0, at time τ1+τ2, illustrating the rate of convergence in a special case of theorem 3.2.

n	areas of the three peaks	heights of the three peaks	mean displacement
10	(0.0279285,0.716249,0.255823)	(0.0733035,1.88015,0.669941)	0.791225
20	(0.0309972,0.725965,0.243038)	(0.0931706,2.18234,0.728788)	0.713194
30	(0.0318689,0.728297,0.239835)	(0.102331,2.33873,0.768103)	0.696690
40	(0.0322853,0.729350,0.238365)	(0.107491,2.42843,0.791417)	0.689617
50	(0.0325301,0.729952,0.237518)	(0.110785,2.48602,0.806553)	0.685678
60	(0.0326914,0.730343,0.236965)	(0.113066,2.52599,0.817114)	0.683162
70	(0.0328059,0.730618,0.236576)	(0.114737,2.55531,0.824886)	0.681414
80	(0.0328913,0.730821,0.236288)	(0.116014,2.57774,0.830840)	0.680129
90	(0.0329575,0.730978,0.236065)	(0.117021,2.59543,0.835543)	0.679144
100	(0.0330104,0.731102,0.235888)	(0.117836,2.60974,0.839352)	0.678364
110	(0.0330535,0.731203,0.235743)	(0.118508,2.62156,0.842498)	0.677731
120	(0.0330894,0.731287,0.235623)	(0.119073,2.63148,0.845140)	0.677208
130	(0.0331197,0.731358,0.235522)	(0.119553,2.63993,0.847391)	0.676768
140	(0.0331457,0.731419,0.235435)	(0.119967,2.64720,0.849331)	0.676392
150	(0.0331681,0.731471,0.235361)	(0.120327,2.65354,0.851020)	0.676068
160	(0.0331878,0.731517,0.235295)	(0.120644,2.65910,0.852504)	0.675785
170	(0.0332051,0.731557,0.235238)	(0.120924,2.66403,0.853818)	0.675537
180	(0.0332205,0.731593,0.235187)	(0.121174,2.66842,0.854990)	0.675316
190	(0.0332343,0.731625,0.235141)	(0.121398,2.67236,0.856041)	0.675120
200	(0.0332467,0.731653,0.235100)	(0.121600,2.67592,0.856990)	0.674943

Density of the flashing Brownian ratchet at time τ1+τ2, starting at stationarity

By properties of diffusion processes with constant diffusion and gradient drift, the Brownian ratchet has a reversible invariant measure π of the form 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).

We use a different characterization of the Brownian ratchet. We take , the domain of , to be the space of real-valued C1 functions f on R with limits at such that f′ is absolutely continuous and has a right derivative, denoted by f′′, with 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 [22, p. 25] and Theorem II.1 showed that generates a Feller semigroup on . Because both boundaries are natural, the Feller semigroup can be restricted to , the subspace of continuous functions vanishing at . Moreover, the subspace of consisting of functions with compact support is a core for the generator. To confirm the first assertion, for every 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 with compact support, as required.

For the second assertion, we take to be the space of real-valued C1 functions f on the circle [0,L) such that f′ is absolutely continuous and has a right derivative, denoted by f′′, with continuous on the circle [0,L). Thus, f(0)=f(L−), f′(0)=f′(L−), Finally, for every , as in (5.2) except with replaced by and sums over n replaced by their n=0 terms. ▪

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 since V (L)=V (0)=0. 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 , and the wrapped flashing Brownian ratchet at time t, starting from x∈[0,L) at time 0, by . Then the one-step transition function for a continuous-state Markov chain has a stationary distribution . (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 displacement of the flashing Brownian ratchet over the time interval [0,τ1+τ2], starting from the stationary distribution , namely 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 between and its image under the mapping The advantage of modifying in this way is that, when regarded as a measure on R, it becomes unimodal instead of U-shaped.

We propose to approximate 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 distribution 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 where 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 n2(τ1+τ2) is even, in which case we replace the latter quantity by n2(τ1+τ2)+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 of with support [−3,1) instead of [0,4). Starting from the approximate at time 0, the second and third panels show the approximations to the density at times τ1 and τ1+τ2, respectively. Figure 5 can be regarded as a more accurate version of figures 1 and 2. Computations show that which is slightly larger than the corresponding number in table 1.

Figure 5.

We mathematically model the full figure 1. Starting from the stationary distribution with support [−3,1), a Brownian motion runs for time τ1=2.4. Then, starting from where the Brownian motion ended, a Brownian ratchet with , L=4, γ=3λ/8 and λ=5, runs for time τ2=2.4. The black curves in all three panels are based on the random walk approximation with n=100. 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].

Another problem that we hope to address in the near future is to establish a strong law of large numbers for flashing Brownian ratchet increments, perhaps analogous to our earlier strong law of large numbers [15] for the sequence of Parrondo-game profits.

Finally, because the evaluation in (5.4) is computationally intensive, it would be a challenging numerical optimization problem to determine the values of τ1 and τ2 that maximize the long-term mean displacement per unit time, . They would depend on α, L and γ.