Article pubs.acs.org/JPCB
Diffusion in a Crowded, Rearranging Environment Rohit Jain and Kizhakeyil L. Sebastian* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India ABSTRACT: It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))2⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))2⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.
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INTRODUCTION Einstein formulated his celebrated theory of Brownian motion in 1905, considering the motion of a particle in an environment of small, fast moving solvent molecules. The fluctuating force exerted by these solvent molecules causes the particle to essentially execute a random walk. In formulating the theory, the time scale of motion of the particle is assumed to be much larger than that of the solvent molecules, causing a complete separation of the time scales. Further, the properties of the environment do not change. This leads to two results: (1) the mean square displacement of the particle is proportional to the time elapsed (this is referred to as Fickian diffusion) and (2) the spatial distribution of position is a Gaussian, this being a result of the central limit theorem.1 Recent observations of diffusion, in which the environment undergoes structural rearrangement at a time scale comparable to that of the time period for which the diffusing particles are observed, have still found the first result, viz., Fickian diffusion, to be valid. Modern imaging techniques have allowed the investigation of the spatial distribution, and surprisingly, in many cases, it is found to deviate from being Gaussian.2 For example, in the diffusive motion of colloids on a bilayer phospholipid tubule as well as in a network of biofilaments, it is repeatedly found that the spatial distribution of the diffusing particle is non-Gaussian, though the diffusion itself is Fickian,2−4 i.e., ⟨(x(T))2⟩ ∝ T. In some cases, specific mechanisms like Coulombic friction5 or mixing by microswimmers have been suggested to be responsible.4,6 However, there are many cases where the behavior seems to be generic and a large body of evidence for generality of nonGaussian distributions in physicochemical and socio-economic processes is now available. The processes are hard sphere diffusion in colloidal suspensions,7−9 liposome diffusion in entangled actin,3 in simulations of 2D colloids10 and hard sphere fluids,11 diffusive motion in supercooled liquids,12 far from equilibrium situations in granular gases and plasmas and © 2016 American Chemical Society
turbulent systems, in systems close to glass and jamming transitions,13,14 diffusion of a tracer particle in a medium of larger spheres9 or long chain molecules,15 price dynamics,16 and stock returns.17 We now summarize the major experimental observations. More details may be found in the nice Nature Materials commentary by Granick et al.3 There are a lot of phenomena in which diffusion is anomalous, i.e., where the probability distribution function is not Gaussian, but are stable distributions.18−20 In these, the distributions have the functional form f(|x|/Tα/2) and the width of the distribution increases like Tα/2 with α ≠ 1. In comparison, normal Brownian motion would lead to a Gaussian distribution with α = 1. In the past, it has been assumed that, if α ≠ 1, then the distribution would be non-Gaussian. In many diffusion processes where the medium rearranges on a time scale less than the time scale of observation of the diffusion, non-Gaussian distributions have been realized. These exhibit a crossover to Gaussian distribution at large enough times. The most interesting aspect is that ⟨(x(T))2⟩ ∝ T at all times irrespective of the form of the distribution. Chubynsky and Slater21 have analyzed the “diffusing diffusivity” model, in which the diffusion coefficient changes with time, due to the rearrangement of the environment. Assuming the infinite time limit of the distribution of the diffusion coefficient to be exponential, they showed analytically that (1) the diffusion is Fickian and (2) it leads to an exponential distribution in position space, whose width increases as T . For larger T, they did simulations and showed that in the limit of large T the distribution is Gaussian, while, at intermediate times, it is in between these two. However, no analytic results were given in the intermediate to Received: February 14, 2016 Revised: March 21, 2016 Published: March 30, 2016 3988
DOI: 10.1021/acs.jpcb.6b01527 J. Phys. Chem. B 2016, 120, 3988−3992
Article
The Journal of Physical Chemistry B large time limit. In this paper, we present a model which can be analytically solved and is valid in all of these limits. The model studied by Chubynsky and Slater21 is simple. A particle undergoes diffusion in an environment that rearranges. This time scale is larger than the time required for the system to have a well-defined diffusion coefficient. Hence, the rearrangement causes changes in the diffusion coefficientit becomes a function of time. Its time evolution is taken to be stochastic, and the corresponding probability distribution function π(D, t) is assumed to obey a diffusion equation of the form ⎫ ∂π (D , t ) ⎧ ∂ 2 ∂ = ⎨F 2 + s(D)⎬π (D , t ) ∂t ∂D ⎩ ∂D ⎭
Using eq 2 in eq 5, one writes the probability density functional for Brownian paths x(t) as 7[x(t )] =
P(xf , T |0, 0) = T
e−i ∫0 (1)
T
⟨ei ∫0
T
⟩ = e−D ∫0
dtp2 (t )
T T dtη(t )p(t )−D ∫0 dtp2 (t ) 0
T
dtη(t )p(t )
⟩ = e− ∫0
P(xf , T ) = T
⟨e− ∫0
dtD(t )p2 (t )
(8)
x(T ) = xf
∫x(0)=0
Dx(t )
∫ Dp(t )
T
dtD(t )p2 (t ) + i ∫0 dtp(t )x(̇ t )
⟩D
(9)
where ⟨...⟩D stands for averaging over all realizations of D(t). The diffusion coefficient D(t) is a stochastic function of time, that obviously is constrained to be positive. Hence, we consider a model in which we take
(3)
D(t ) = ξ 2(t )
(10)
where ξ = {ξ1(t), ξ2(t), ..., ξn(t)} is an n-dimensional random vector. A simple, analytically solvable model would result if we take ξ(t) to be the position vector of an n-dimensional harmonic oscillator that undergoes overdamped Brownian motion. The frequency associated with all the directions may be taken to be the same, which we denote as ω. The “diffusion coefficient”, F, too is taken as the same. Then, the averaging over all D(t) in eq 9 is conveniently expressed as a phase space path integral,22,23 and one can write the probability distribution function for the combined system (xf, ξf) as
(4)
The above equation means that the probability distribution functional for η(t) may be written as the Fourier transform
∫ Dp(t )e−i ∫
(7)
Clearly, T ≫ tc and is typically of the order of tR or larger than it. When the environment fluctuates as a function of time, which leads to the effective time dependent diffusion coefficient D(t), one can generalize the above expression to
The symbol ⟨....⟩ stands for averaging over the noise η(t). The best way to characterize the noise η(t) is through its characteristic functional
7̃ [η(t )] =
T
dtx(̇ t )p(t ) − D ∫0 dtp2 (t )
(2)
dtη(t )p(t )
∫ Dp(t )
BROWNIAN MOTION WITH STOCHASTIC DIFFUSION COEFFICIENT We now consider a situation where the environment rearranges on a time scale tR. Also, let us say that we are observing diffusion over a time tobs ≥ tR. tR is long enough such that it is possible to define a diffusion coefficient for the particle in that time interval, and typically one would have D(t) = ⟨(Δx)2⟩/ (2Δt), where Δx is the displacement of the particle during an interval Δt centered around the time t, satisfying Δt ≫ tc. Further, it is also necessary that tR ≫ tc. In such a case, the characteristic functional of the noise will be
which represents overdamped motion of the particle. η(t) is the noise acting on the particle, arising from the random collisions of solvent molecules with the diffusing particle. It is assumed to have a very short correlation time tc in comparison with the time scale tB at which the motion of the Brownian particle takes place, i.e., tB ≫ tc. In this limit, it is appropriate to assume that η(t) is a Gaussian white noise having the correlation function
T
Dx(t )
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PHASE SPACE PATH INTEGRALS FOR BROWNIAN MOTION Here we give a model for diffusing diffusivity, which can be solved analytically. We shall use phase space path integrals22 in our analysis, as that leads to the solution straightforwardly. The position x(t) of a Brownian particle undergoing diffusion is governed by the stochastic differential equation
⟨ei ∫0
x(T ) = xf
∫x(0)=0
where the path integral is over all paths in the phase space (see the book by Kamenev22 for a discussion of phase space path integrals and the paper by Janakiraman and Sebastian23 for more details), with the initial and final conditions of position specified, while the momenta can have any value in the range (−∞, ∞). A rigorous definition of the integrals in eq 7 is given in Appendix A.
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⟨η(t )η(t ′)⟩ = 2Dδ(t − t ′)
(6)
(see Appendix A for more details). Hence, the probability of finding a particle at xf at time T given that it started at xi = 0 at time t = 0 is given by
As D has to be greater than zero, it diffuses only in the halfspace with D > 0. Hence, one imposes reflecting boundary conditions at D = 0. At short times (i.e., t < τD, with τD being the characteristic time for the diffusion of diffusivity), the diffusivity of the particle can be assumed constant. Therefore, for an ensemble of particles, the distribution of displacements for small times does not depend on the diffusion of diffusivity but only the distribution of diffusivity which may be assumed to have its equilibrium value, πe(D). Taking πe(D) = exp(−D/ D0)/D0, with D0 being a constant, it is easy to show that the probability distribution for the displacement x(T) is exponential, with a width proportional to T (Fickian diffusion).21 For larger values of T, taking F and s(D) both to be constants and taking the diffusion to be confined to the interval [0, ∞), simulations with time evolving diffusion coefficients were done, and the result was a crossover to Gaussian distribution at large times.
x(̇ t ) = η(t )
T T dtx(̇ t )p(t )−D ∫0 dtp2 (t ) 0
∫ Dp(t )e−i ∫
(5) 3989
DOI: 10.1021/acs.jpcb.6b01527 J. Phys. Chem. B 2016, 120, 3988−3992
Article
The Journal of Physical Chemistry B x(T ) = xf
∫x(0)=0
P(xf , ξf , T |0, ξ0 , 0) =
where
Dx(t )
ξ(T ) = ξf
P jsink(ξjf , T |ξj0 , 0; pT ) =
∫ Dp(t )e
T i ∫0 dtp(t )x(̇ t )
∫
− ∫0 dt {ξ 2(t )p2 (t ) + Fπ 2(t ) + iπ(t )·(ξ(̇ t ) + ωξ(t ))}
∫ξ(0)=ξ
Dξ(t )
0
exp[−
T
Dπ(t )e
(11)
ω F
x , p̃ =
F ω
ω ξ, F j
p, ξj̃ =
F ω
and πj̃ =
T
j
Dξj(t )
∫ Dπj(t )
j0
dt {πj 2(t ) + iπj(t )(ξj(̇ t ) + ξj(t ))
+ pT 2 ξj 2(t )}]
where π(t) = (π1(t), π2(t), ..., πn(t)), with πj(t) being the conjugate momentum associated with ξj(t). The paths ξj(t) satisfy the boundary conditions ξj(0) = ξj0 and ξj(T) = ξjf. The probability distribution for the Brownian particle alone can be obtained from P(xf, ξf, T|0, ξ0, 0) by averaging over the initial probability distribution Pini(ξ0) and integrating over all possible final positions of ξf. At this point, we introduce the ω dimensionless quantities defined by t ̃ = ωt, D̃ = F D, x̃ =
∫0
ξj(T ) = ξjf
∫ξ (0)=ξ
(17)
Interestingly, Psink j (ξjf, T|ξj0, 0; pT) is the probability of finding the jth Brownian oscillator at ξjf in the presence of a sink which absorbs it at the position ξj at a rate equal to pT2ξj2, given that it started at ξj0 at time t = 0. Thus, one model of dif f using dif f usivity has been reduced to the problem of Brownian motion with absorption in n-dimensional space24,25 (see Figure 1), a problem for which an exact solution is easy to find.
πj . Writing eq
11 in terms of these dimensionless quantities and dropping the “tildes” to simplify the notation, we get x(T ) = xf
∫x(0)=0
P(xf , ξf , T |0, ξ0 , 0) = T dtp(t )x(̇ t ) 0
∫ Dp(t )ei ∫
Dx(t )
ξ(T ) = ξf
∫ξ(0)=ξ
Dξ(t )
∫ Dπ(t )
0
exp[−
∫0
T
dt {ξ 2(t )p2 (t ) + π 2(t ) + iπ(t ) ·
(ξ(̇ t ) + ξ(t ))}]
(12)
Integrating the term ∫ T0 dtp(t)ẋ(t) by parts and noting that x(0) = 0 and then doing the path integral over x(t) (see the paper by Janakiraman and Sebastian23), one gets P(xf , ξf , T |ξ0 , 0) = ξ(T ) = ξf
∫ξ(0)=ξ
Dξ(t )
T dtp ̇(t )x(t ) + ip(T )xf 0
∫ Dp(t )e−i ∫
Figure 1. If one takes n = 2, then the problem of diffusing diffusivity is equivalent to the problem of a “fictitious” particle (shown in green color) diffusing in a two-dimensional plane, with absorption. Its 1 position coordinates are (ξ1, ξ2), and V(ξ1, ξ2) = 2 (ξ12 + ξ2 2) is the potential it feels.
δ[p ̇(t )]
∫ Dπ(t )
0
exp[−
∫0
T
2
2
As the action of the path integral in eq 17 is quadratic, it can be easily evaluated. Another possibility is to use the fact that Psink j (ξjf, T|ξj0, 0; pT) represents the propagator for Brownian motion of a harmonic oscillator in the overdamped limit, in the presence of a sink which absorbs it at a rate pT2ξj2.24,25 Hence, it obeys the Smoluchowski equation
2
dt {ξ (t )p (t ) + π (t ) + iπ(t ) ·
(ξ(̇ t ) + ξ(t ))}]
(13)
(see Appendix A for more details). In the above, δ[...] is the Dirac delta functional. δ[ṗ(t)] implies p ̇(t ) = 0 for all t
⎛ ⎞ 2 ⎜ ∂ − ∂ − ∂ ξj + p 2 ξj 2⎟P jsink(ξj , T |ξj0 , 0; p ) = 0 ⎜ ∂t ⎟ 2 T T ∂ξj ∂ξj ⎝ ⎠
(14)
Noting that the solution of the above equation is p(t) = pT, where pT is a constant and is equal to p(T) and p(0), and putting in the normalization factor, we get ∞
P(xf , ξf , T |ξ0 , 0) =
∫−∞
∫ Dπ(t ) exp[−∫0
T
dpT 2π
e
ipT xf
ξ(T ) = ξf
∫ξ(0)=ξ
(18)
with the initial condition P jsink(ξj , 0|ξj0 , 0; pT ) = δ(ξj − ξj0)
Dξ(t )
This can be solved to get
0
P jsink(ξjf , T |ξj0 , 0; pT )
dt {ξ 2(t )pT 2 + π 2(t )
+ iπ(t ) ·(ξ(̇ t ) + ξ(t ))}]
P(xf , ξf , T |ξ0 , 0) =
∫−∞
dpT 2π
α
=
(15)
2 π sinh(αT )
This may be rewritten as ∞
(19)
e(T /2) − (ξjf
2
/4) + (ξj0 2 /4)
1 exp − α csch(αT )((ξjf 2 + ξj0 2) cosh(αT ) 4
{
n
eipT xf (∏ P jsink(ξjf , T |ξj0 ; pT ))
− 2ξjf ξj0)
j=1
(16) 3990
}
(20) DOI: 10.1021/acs.jpcb.6b01527 J. Phys. Chem. B 2016, 120, 3988−3992
Article
The Journal of Physical Chemistry B α=
1 + 4pT 2
⎞ ∂π ∂ ⎛⎜ ∂π = + 2ωDπ ⎟ 4FD ⎠ ∂t ∂D ⎝ ∂D
(21)
We take the initial distribution of the Brownian oscillator to be the equilibrium one, given by Pj ,eq(ξj0) =
⎛ ξ 2⎞ 1 j0 ⎟ exp⎜⎜ − ⎟ 2π ⎝ 2 ⎠
This means that the flux is given by ⎛ ⎞ ∂π + 2ωDπ ⎟ j = −⎜4FD ⎝ ⎠ ∂D
(22)
As we are interested only in the final position xf of the particle, we integrate over all possible final positions of the Brownian oscillator. Thus, we have ∞
P(xf , T ) =
∫−∞
dpT 2π
n
e
∏ (∫
ipT xf
j=1
∞
−∞
0
∫−∞ dξj0Pj,eq(ξj0)
P jsink(ξjf , T |ξj0 , 0; pT ))
model of Chubynsky and Slater.21 Therefore, even though the plots in Figure 2 strikingly resemble the simulation results of Chubynsky and Slater21 at all times, we have not compared our numerical results with theirs.
(23)
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which can be evaluated to get ∞
P(xf , T ) =
∫−∞
dpT 2π
eipT xf P ̅(pT , T )
CONCLUSIONS One of the diffusing diffusivity models has been shown to be equivalent to Brownian motion with absorption, leading to analytical solutions for the problem. At all times, we show ⟨x(T)2⟩ ∝ T. At short times, the distribution of displacements is found to be exponential, while, at long times, it crosses over to being Gaussian. Our analytical results have striking similarity with the numerical results obtained by Slater et al.21
(24)
with ⎛ ⎞n /2 4α e−(α − 1)T ⎜ ⎟ P ̅(pT , T ) = ⎜ 2 2 − 2α T ⎟ ⎝ (α + 1) − (α − 1) e ⎠
(25)
■
We consider only the case n = 2, as this leads to a probability distribution for D that is exponential in D, which is the model considered by Chubynsky and Slater.21 In this case, we get ⎛ ∂ 2P ̅(p , T ) ⎞ T ⎟⎟ ⟨xf ⟩ = −⎜⎜ ∂pT 2 ⎝ ⎠p 2
T
(27)
which clearly shows that in this model the diffusion coefficient of diffusivity is proportional to D. Further, the steady state solution to this equation with zero flux at D = 0 is given by 1 2F πe(D) = D e−D / D0 , with D0 = ω , which is the same as in the
∞
dξjf
(26)
APPENDIX A
Phase Space Path Integrals for Brownian Motion
We give below the details regarding the formalism and refer the reader to the book by Kamenev22 and the paper by Janakiraman and Sebastian23 for more details. We divide the time interval (0, T) to N equal intervals each of duration Δt such that NΔt = T and use the notation xj = x(jΔt) with j = 0, 1, 2, ..., N, x0 = xi, and xN = xf. Eventually, we shall take the limit N → ∞ and Δt → 0 such that NΔt = T. The differential eq 2 can be discretized as
= 4T =0
which shows that, for any T, diffusion is Fickian. Further, in the small T limit (αT ≪ 1, which implies T ≪ 1 and pT ≪ 1/(2 T ) or T ≪ 1 and xf ≫ 2 T ), P(xf, T) has exponential form. Note that this does not happen if T > 1. On the other hand, if T ≫ 1, P̅(pT, T) ≈ e−T(α−1), which 2 behaves like e−2pT T for small pT and like e−2T|pT| for large values of pT. This means that, for large xf, the distribution is Gaussian, while, for small xf, it resembles a Cauchy distribution. In Figure 2, we present the numerical evaluation of eq 24. At short times, the distribution is exponential, while, at large times, it becomes Gaussian. We note that, with n = 2 and D = ξ12 + ξ22, the Fokker−Planck equation that D obeys is not identical with eq 1 but is given by
xj − xj − 1 = ηj
(28)
where ηj = ∫ jΔt (j−1)Δtdtη(t). Clearly, ⟨ηjηi⟩ = 2DδijΔt. The 2
characteristic function for ηj is ⟨eipjηj⟩ = e−DΔtpj , which means that the probability distribution of ηj is given by Pj̃(ηj) =
1 2π
∞
2
∫−∞ dpe−Dp Δt−ip η j
j j
(29)
Hence, the probability distribution for the variables {η1, η2, ..., ηN} is given by P(̃ η1 , η2 , ..., ηN ) =
⎛ 1 ⎞N ⎜ ⎟ ⎝ 2π ⎠ ∞
...
∞
∞
∫−∞ dp1 ∫−∞ dp2
∫−∞ dpN e− ∑
N (Dpj 2 Δt + ipj ηj) j=1
(30)
This is the precise definition of the path integral which was written as eq 5. Now we can change variables from {η1, η2, ..., ηn} to {x1, x2, ..., xN} (note that xN = xf and that the Jacobian of the transformation is unity) and write the probability density for (x1, x2, ..., xN) as
Figure 2. Probability distribution function in terms of dimensionless variables, for different values of time. 3991
DOI: 10.1021/acs.jpcb.6b01527 J. Phys. Chem. B 2016, 120, 3988−3992
Article
The Journal of Physical Chemistry B 7(x1 , x 2 , ..., xN ) =
⎛ 1 ⎞N ⎜ ⎟ ⎝ 2π ⎠
∞
∞
...
Swimming Eukaryotic Microorganisms. Phys. Rev. Lett. 2009, 103, 198103. (5) Menzel, A. M.; Goldenfeld, N. Effect of Coulombic friction on spatial displacement statistics. Phys. Rev. E 2011, 84, 011122. (6) Kurtuldu, H.; Guasto, J. S.; Johnson, K. A.; Gollub, J. P. Enhancement of biomixing by swimming algal cells in two-dimensional films. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 10391−10395. (7) Kegel, W. Direct Observation of Dynamical Heterogeneities in Colloidal Hard-Sphere Suspensions. Science 2000, 287, 290−293. (8) Guan, J.; Wang, B.; Granick, S. Even Hard-Sphere Colloidal Suspensions Display Fickian Yet Non-Gaussian Diffusion. ACS Nano 2014, 8, 3331−3336. (9) Kwon, G.; Sung, B. J.; Yethiraj, A. Dynamics in Crowded Environments: Is Non-Gaussian Brownian Diffusion Normal? J. Phys. Chem. B 2014, 118, 8128−8134. (10) Kim, J.; Kim, C.; Sung, B. J. Simulation Study of Seemingly Fickian but Heterogeneous Dynamics of Two Dimensional Colloids. Phys. Rev. Lett. 2013, 110, 047801. (11) Saltzman, E. J.; Schweizer, K. S. Large-amplitude jumps and non-Gaussian dynamics in highly concentrated hard sphere fluids. Physical review. E, Statistical, nonlinear, and soft matter physics 2008, 77, 051504. (12) Eaves, J. D.; Reichman, D. R. Spatial dimension and the dynamics of supercooled liquids. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 15171−15175. (13) Weeks, E. R. Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition. Science 2000, 287, 627−631. (14) Chaudhuri, P.; Berthier, L.; Kob, W. Universal nature of particle displacements close to glass and jamming transitions. Phys. Rev. Lett. 2007, 99, 060604. (15) Chakrabarti, R.; Kesselheim, S.; Košovan, P.; Holm, C. Tracer diffusion in a crowded cylindrical channel. Phys. Rev. E 2013, 87, 062709. (16) Majumder, S.; Diermeier, D.; Rietz, T. Price dynamics in political prediction markets. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 679−684. (17) Silva, A. C.; Prange, R. E.; Yakovenko, V. M. Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact. Phys. A 2004, 344, 227−235. (18) Hofling, F.; Franosch, T. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 2013, 76, 046602. (19) Metzler, R.; Jeon, J.; Cherstvy, A. G.; Barkai, E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 2014, 16, 24128. (20) Sokolov, I. M. Models of anomalous diffusion in crowded environments. Soft Matter 2012, 8, 9043−9052. (21) Chubynsky, M. V.; Slater, G. W. Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion. Phys. Rev. Lett. 2014, 113, 098302. (22) Kamenev, A. Field Theory of Non-equilibrium Systems; Cambridge University Press: New York, 2011. (23) Janakiraman, D.; Sebastian, K. Path-integral formulation for Lévy flights: Evaluation of the propagator for free, linear, and harmonic potentials in the over-and underdamped limits. Phys. Rev. E 2012, 86, 061105. (24) Bagchi, B.; Fleming, G. Dynamics of Activationless Reactions in Solution. J. Phys. Chem. 1990, 94, 9−20. (25) Sebastian, K. L. Theory of electronic relaxation in solution: Exact solution for a δ-function sink in a parabolic potential. Phys. Rev. A: At., Mol., Opt. Phys. 1992, 46, R1732.
∞
∫−∞ dp1 ∫−∞ dp2
∫−∞ dpN e− ∑
N (Dpj 2 Δt + ipj (xj − xj − 1)) j=1
(31)
This is the correct, discretized version of eq 6 in the paper. To find the probability distribution P(xf, T|xi, 0), one has to integrate over all possible values of {x1, x2, ..., xN−1}, leading to ⎛ 1 ⎞N ⎜ ⎟ N →∞⎝ 2π ⎠
P(xf , T |xi , 0) = lim
Δt → 0 ∞
∞
...
∞
∞
∫−∞ dx1 ∫−∞ dx2
∞
∞
∫−∞ dxN−1∫−∞ dp1 ∫−∞ dp2 ... ∫−∞ dpN N
e− ∑ j=1(Dpj
2
Δt + ipj (xj − xj − 1))
(32)
This is the discretized version of the phase space path integral, given in eq 7. If we perform integrals over all pj, then one gets the usual position space path integral describing Brownian motion, which is usually used in textbooks. We now illustrate our procedure in going from eq 12 to eq 15 by applying it to the evaluation of the above integral. An easy way to evaluate the path integral is to perform the integral over all x1, x2, ..., xN−1 which results in ⎛ 1 ⎞N ⎜ ⎟ N →∞⎝ 2π ⎠
P(xf , T |xi , 0) = lim
Δt → 0
N
e−i(pN xN − p1x0)e− ∑ j = 1 Dpj
2
∞
∞
∞
∫−∞ dp1 ∫−∞ dp2 ... ∫−∞ dpN N
Δt
(2π )N − 1 ∏ δ(pj − pj − 1 ) j=2
(33)
On performing the integrals over {p1, p2, ..., pN−1}, to remove all the delta functions, we get ∞
P(xf , T |xi , 0) = =
■
∫−∞
dpN
e−ipN (xN − x0)e−D(N Δt )pN
2
2π 2 1 e−(xf − xi) /4DT 4πDT
(34)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +918022932385. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Professors Ralf Metzler and Arindam Chowdhury for useful discussions. The work of K.L.S. was supported by the J C Bose Fellowship of the Department of Science and Technology, Government of India.
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REFERENCES
(1) Frey, E.; Kroy, K. Brownian motion: a paradigm of soft matter and biological physics. Ann. Phys. 2005, 14, 20−50. (2) Wang, B.; Anthony, S. M.; Bae, S. C.; Granick, S. Anomalous yet Brownian. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 15160−15164. (3) Wang, B.; Kuo, J.; Bae, S.; Granick, S. When Brownian diffusion is not Gaussian. Nat. Mater. 2012, 11, 481. (4) Leptos, K. C.; Guasto, J. S.; Gollub, J.; Pesci, A. I.; Goldstein, R. E. Dynamics of Enhanced Tracer Diffusion in Suspensions of 3992
DOI: 10.1021/acs.jpcb.6b01527 J. Phys. Chem. B 2016, 120, 3988−3992
