Normal and Anomalous Diffusion: An Analytical Study Based on

Article pubs.acs.org/JPCB

Normal and Anomalous Diffusion: An Analytical Study Based on Quantum Collision Dynamics and Boltzmann Transport Theory Sathiya Mahakrishnan, Subrata Chakraborty, and Amrendra Vijay* Department of Chemistry, Indian Institute of Technology Madras, Chennai 600036, India S Supporting Information *

ABSTRACT: Diffusion, an emergent nonequilibrium transport phenomenon, is a nontrivial manifestation of the correlation between the microscopic dynamics of individual molecules and their statistical behavior observed in experiments. We present a thorough investigation of this viewpoint using the mathematical tools of quantum scattering, within the framework of Boltzmann transport theory. In particular, we ask: (a) How and when does a normal diffusive transport become anomalous? (b) What physical attribute of the system is conceptually useful to faithfully rationalize large variations in the coefficient of normal diffusion, observed particularly within the dynamical environment of biological cells? To characterize the diffusive transport, we introduce, analogous to continuous phase transitions, the curvature of the mean square displacement as an order parameter and use the notion of quantum scattering length, which measures the effective interactions between the diffusing molecules and the surrounding, to define a tuning variable, η. We show that the curvature signature conveniently differentiates the normal diffusion regime from the superdiffusion and subdiffusion regimes and the critical point, η = ηc, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example) as a dynamical cause of the perpetual molecular movement, which eventually manifests as an ordered motion, called the diffusion. diffusive transport, through the pioneering work of Fick,44 and displays wide numerical variations, particularly among different biological systems. For example, the experimental values of D (in m2/s) for the potassium ion in water,45 Rhodamine 6G (Rh6G) in low-concentration aqueous solutions,33,34 the green fluorescent protein (GFP) in the cytoplasm of Escherichia coli45,46 and the interphase chromatin in the living cells of Saccharomyces cerevisiae45,47 respectively are about 2 × 10−9, 3.5 × 10−10, 7 × 10−12 and 5 × 10−16. What physical attributes of the system are then useful and appropriate to rationalize such order of magnitude differences in the experimentally observed D? We here address these substantial questions within the framework of Boltzmann transport theory,48−55 in conjunction with quantum collision dynamics among constituent molecular entities.56−58 In particular, we show that the dynamical

I. INTRODUCTION Complex dynamical processes in biological, soft matter, and other physical systems are often characterized by the mean square displacement (MSD), ⟨r·⃗ r⟩⃗ − ⟨r⟩⃗ ·⟨r⟩⃗ , which, for normal diffusive transport, increases linearly with time  a fact wellknown through the pioneering works of Einstein, Smoluchowski and others.1−6 Deviations from this linear law, often termed as the anomalous diffusion, have attracted much attention in recent years as they are found to be ubiquitous,7−32 particularly within the crowded environment of biological cells.7−16,32−35 Diffusion is axiomatically an emergent nonequilibrium statistical phenomenon, the chief dynamical cause of which, inter alia, is the microscopic collisions among constituent molecular entities within the system. How and when does a normal diffusive transport become anomalous? Related to this question, and equally important, is the coefficient of diffusion, D, which embodies the dynamical response arising due to the concentration (equivalently, the molecular density) gradient present within the system,33−43 defines the notion of normal © 2016 American Chemical Society

Received: June 23, 2016 Revised: August 21, 2016 Published: August 23, 2016 9608

DOI: 10.1021/acs.jpcb.6b06380 J. Phys. Chem. B 2016, 120, 9608−9620

Article

The Journal of Physical Chemistry B

approximation for Ĝ ab , which, for a rapid numerical convergence in a given situation, is judiciously selected based on physical considerations. For example, the local thermal equilibrium, for being a unique physical state, is often a convenient reference configuration to obtain an initial estimate ⃗ For practical computations, now, there are a number for f(r,⃗ k,t). of algorithms such as the singular perturbation theory,71−75 moment expansion52,53,74,76 and the model kinetic theory,74,77 which aim to implement the (physically motivated) process of linearization, and they all differ essentially in their choice of approximation for Ĝ ab. Quite often, we note, the first nontrivial approximation for f(r,⃗ k⃗,t) is sufficient to rationalize the emergent nonequilibrium phenomenon of interest. Historically, the singular perturbation theory, following the pioneering works of Hilbert,71 Chapman,72,73 Enskog,54 and others,54,74,75 represents the first analytically successful approximation framework for the Boltzmann equation. Within this framework, one identifies the dimensionless Knudsen number (Kn) as a small parameter to obtain a successive (infinite) series approximation to the Boltzmann equation. For each order in Kn, one then has an appropriately parametrized linear integral equation, to be solved analytically or by numerical means.51,74 The philosophy of model kinetic theory, on the other hand, is to introduce an effective collision operator in place of the Boltzmann collision term and obtain an appropriately para⃗ a metrized differential equation (linear or nonlinear) for f(r,⃗ k,t), notable example of which is the model due to Bhatnagar, Gross and Krook74,77 and its variants, including a successful application on electron conduction problem by Lindhard.78 Finally, a definite example of the method of moment expansion is the work originally due to Grad,52,53,76,79 who expressed ⃗ in terms of Hermite polynomials and thereby introduced f(r,⃗ k,t) a 13-moment approximation to the Boltzmann equation. Strengths and weaknesses of these and other related methods are well-summarized in the literature.48,51,54,74 Next, a critical input for constructing the collision operator, Ĝ ab, is the scattering amplitude, Sab(kR⃗ ,kR′⃗ ), the explicit mathematical form of which, inter alia, determines the mathematical complexities associated with the Boltzmann equation. Earlier studies on this subject have mostly used classical mechanical ideas to obtain Sab(kR⃗ ,kR′⃗ ), which may not always be appropriate. In fact, the use of quantum mechanical expressions for Sab(kR⃗ ,kR′⃗ ) in the Boltzmann equation, for practical studies, have mostly remained a less explored area.80 An objective of the present study is to fill this significant gap. Notably, there are a number of well-developed quantum wave packet-based molecular scattering tools,58,81 which may be adapted for the purpose. In the present work, we are focused on the completely analytical solution of the Boltzmann equation. To achieve the objective, here, we combine the basic philosophy of model kinetic theory with Grad’s idea of Hermite series expansion52,53,79 and incorporate a number of significantly new ideas for the final implementation of the transport theory. To be specific, we introduce a quantum mechanical framework for Sab(kR⃗ ,k⃗R′ ) to construct the collision operator, Ĝ ab, and subsequently obtain a general form of the effective linear collision operator, useful for a variety of practical transport studies. As we will see presently, the coefficient of normal diffusion, D, may be conveniently expressed in terms of quantum scattering length, l(ab) scat , which is a fundamental physical attribute of the interacting system, and therefore l(ab) scat may be fruitfully used to rationalize emergent diffusive transport processes in complex systems. Finally, we note that the

transition from normal to the anomalous diffusion regime may be fruitfully viewed as analogous to the phenomenon of continuous phase transition in magnetic materials and liquid− gas systems. To complete the analogy, following the classic works of Landau and others,59,60 we here introduce the slope, μ(t) = d[⟨r·⃗ r⟩⃗ − ⟨r⟩⃗ ·⟨r⟩⃗ ]/dt, and the curvature, ψ = dμ(t)/dt, as appropriate order parameters, along with a tuning variable, η, which, at a given temperature, is related to the quantum scattering length, l(ab) scat , encoding the complete information on dynamical interactions present in the system. The curvature signature (ψ = positive, zero and negative) conveniently differentiates the normal (ψ = zero) diffusion regime from the superdiffusion (ψ = positive) and subdiffusion (ψ = negative) regimes. Furthermore, the critical point, η = ηc, that separates the normal and anomalous diffusion regimes, in essence, measures the deviation of the initial molecular distribution function from the local thermal equilibrium one, which, in turn, also furnishes the complete information on the coefficient of normal diffusion. Next, as we will see presently, the scattering length, l(ab) scat , which essentially measures the effective interaction, for low energy collisions, between the diffusing molecules and the surrounding molecular species, is a useful and well-defined physical attribute to rationalize the wide variations in D observed in experiments. In what follows, we present a brief outline of key analytical approaches to the Boltzmann equation available in the literature vis a vis the mathematical method used in the present study. Why is the Boltzmann transport theory an appropriate mathematical paradigm to study the diffusive transport processes? First of all, the framework of the Boltzmann equation, as a conceptually unifying basis to rationalize a wide variety of nonequilibrium statistical mechanical phenomena in terms of microscopic scattering events among constituent molecular entities and thus establishing an unambiguous connection between the microscopic and the emergent world, has a rich history, well-summarized in the literature.48−54,61−67 Even today, for example, the pioneering works of Enskog,54 the starting point of which is the Boltzmann equation, stand as a definite and successful mathematical exposition, albeit of prequantum era, on the subject of diffusion and other kinetic processes. Other theoretical approaches such as the Fokker− Planck type of equations,68 though eminently useful, represent only an approximation to the Boltzmann equation,66,69 a critical analysis of which, in the context of Brownian motion, has been presented by Keilson and Storer.66 In the present work, therefore, we use the Boltzmann equation51 as a central theme to study the diffusion processes. As is well-known, the analytical complexities associated with the Boltzmann equation are primarily due to the mathematical form of the nonlinear integral collision operator, Ĝ ab, which encodes the complete information on the microscopic dynamics of constituent molecular entities, in terms of wave vector-dependent scattering probability, Sab(kR⃗ ,kR⃗′ ). First, by definition, Ĝ ab, like the Hartree−Fock potential for interacting electrons, depends ⃗ upon the distribution function, f(r,⃗ k,t), which itself is the solution of the Boltzmann equation one seeks to determine.51,70 Accordingly, the solution of the Boltzmann equation, as a practical necessity, must be implemented as an iterative mathematical process. An iterative process, assuming Sab(kR⃗ ,kR′⃗ ) for molecular scattering events have already been obtained in a suitable mathematical form (either by classical or a quantum mechanical means), requires a nontrivial initial choice for ⃗ and this is frequently accomplished by using a linear f(r,⃗ k,t), 9609


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The Journal of Physical Chemistry B contemporary discourse on diffusion (normal as well as anomalous) have mostly used mathematical models that are essentially based on the notion of random walks,7,20,82,83 which may be viewed as complementary to the present work. The paper is organized as follows. In section II, we define the transport equation and the observables of interest. In section III, we present a Hermite series expansion of an arbitrary realvalued and positive-definite one-particle distribution function on the real line. In section IV, we evaluate the Boltzmann collision operator and outline a well-defined sequence of an approximation scheme to obtain an effective linear collision operator, suitable for a variety of nonequilibrium transport studies. In section V, we present a solution of the linear Boltzmann equation and obtain an analytical expression for the observables. In section VI, we discuss the analytical as well as numerical results on diffusion, with representative examples from biological systems. Finally, we conclude the paper and briefly discuss the future outlook in section VII.

=

⎞ ∂ ∑ [ηi , Ĥeff ] − ∑ Gab̂ ⎟⎟fa ( r ⃗ , k ⃗ , t ) = 0 ∂ηi ⎠ i b

∫ ∫ d3r d3k1 x 2 ∂∂t fa ( r ⃗ , k1⃗ , t )

⎡L 2 2 δg 0δg 0⎢ x δg 0 + 2 (1 − δg 0) y z ⎢ 3 x x g ⎣ gx , g y , g z x ⎤ ∂ cos(gx Lx)⎥ d3k1 fa (g ⃗ , k1⃗ , t ) ⎥⎦ ∂t 1 Na

∑

∫

(2)

where 2Lx is the linear dimension of the system. Finally, the curvature ψ is obtained as equal to dμ(t)/dt, using eq 2.

III. TENSOR HERMITE POLYNOMIALS We here describe the expansion of an arbitrary, real-valued and ⃗ on positive-definite one-particle distribution function, f(r,⃗ k,t), the real line, in terms of the complete set of Hermite polynomials (see Supporting Information, section I). Let us consider the eigenfunctions of the Hamiltonian operator for a three-dimensional harmonic oscillator in the Cartesian space (r ⃗ = x̂ x + ŷ y + ẑ z) as follows.

II. TRANSPORT EQUATION AND OBSERVABLES A key mathematical object that unifies various nonequilibrium phenomena in nature is the distribution function fa(r,⃗ k1⃗ ,t), ⃗ = ∫ d3r na(r,⃗ t) = Na. That is, d3r d3k where ∫ ∫ d3r d3k fa(r,⃗ k,t) ⃗ defines the mean number of molecular species a in the fa(r,⃗ k,t) volume element d3r located at r,⃗ whose wave vectors lie in d3k about the wave vector k⃗ at time t. Accordingly, na(r,⃗ t) is the local number density and Na is the total number of molecular ⃗ species a present in the system under consideration. fa(r,⃗ k,t) satisfies the nonlinear integro-differential equation, originally due to Boltzmann,48−51 as given below. ⎛∂ ⎜⎜ + ⎝ ∂t

1 Na

μ(t ) =

χm⃗ ( r ⃗) = χ0 (r )Hm⃗( r ⃗)

(3)

where 1 exp( −r 2/2) π 3/4

χ0 (r ) =

(4)

and Hm⃗( r ⃗) =

(1)

Hmx(x)

Hmy(y)

Hmz(z)

mx

my

mz

1/2

(2 mx ! )

1/2

(2 my ! )

(2 mz ! )1/2

(5)

where m⃗ stands for the triplet of integers: 0 ≤ mx < ∞, 0 ≤ my < ∞, and 0 ≤ mz < ∞. Hmx(x), Hmy(y), and Hmz(z) are the Hermite polynomials. An arbitrary function ψ(r)⃗ may then be expanded as follows.

where (η1, η2, η3) and (η4, η5, η6) respectively are the components of the position r ⃗ and the wave vector k⃗ of the molecular species a. Ĥ eff stands for an effective one-particle Hamiltonian including the external time-dependent field (electromagnetic, for example) and hence it is, in general, a function of time t and ηk. In a classical dynamical description, ℏk⃗ defines the momentum of the molecule and the symbol [...] stands for the Poisson bracket. The quantity ∑b Ĝ ab denotes the nonlinear integral collision operator which encodes the complete dynamical interactions among all molecular species and accounts for the absorption and emission of molecular species a from a definite wave vector range due to scattering. Evidently, the detailed nature of Ĝ ab is the source of rich physical content and consequent mathematical complexities associated with the Boltzmann equation. We will say more on this later. The observable of interest here is the mean square displacement (MSD) of a molecular species, which is defined as equal to ⟨r·⃗ r⟩⃗ − ⟨r⟩⃗ ·⟨r⟩⃗ . Let us assume that there is no external field acting on the system and the average momentum, at time t = 0, is zero; that is, ⟨ℏk⃗ ⟩ = (1/Na) ∫ ∫ d3r d3k (ℏk)⃗ ⃗ = 0. Notably, this does not necessarily require fa(r,⃗ k,0) ⃗ fa(r,⃗ k,0) to be the distribution function at thermal equilibrium. By construction, then, ⟨ℏk⟩⃗ remains zero at all later times, and therefore (d/dt) ⟨r⟩⃗ 2 = 2 ⟨r⟩⃗ ·(d/dt) ⟨r⟩⃗ = 2 ⟨r⟩⃗ ·⟨ℏk⟩⃗ = 0. In what follows, without loss of generality, we consider the diffusion problem in one dimension and accordingly we compute the slope μ(t) (=d⟨x2 ⟩/dt) as follows.

ψ ( r ⃗) =

∑ Cm⃗χm⃗ ( r ⃗) = χ0 (r) ∑ Cm⃗Hm⃗( r ⃗) → ⎯ m

→ ⎯ m

(6)

We now use eq 6 to construct a positive-definite function ρ(r)⃗ as given below. ρ( r ⃗) = |ψ ( r ⃗)|2 = (χ0 (r ))2

∑ ∑ dmm⃗ ⃗ ′Hm⃗( r ⃗)Hm⃗′( r ⃗) → ⎯ m

→ ⎯ ′ m

(7)

∞

⇒

dmm ∫−∞ d3r ρ( r ⃗) = ∑ ⃗ ⃗ → ⎯

(8)

m

where (Cm⃗ ′) × Cm⃗ = dm⃗ m⃗ ′ = (dm⃗ ′m⃗ ) *. We now change the 1/2 ⃗ variable r ⃗ as follows: r ⃗ = ℏk/(2mk ⇒ r2 = ℏ2k2/(2mkBT) BT) 3 2 3/2 3 ⇒ d k = (2mkBT/ℏ ) d r. Analogous to eq 7, we may then write the positive-definite function in the k-space as follows. ⎡⎛ 2 ⎞1/2 ⎤ ⎢⎜ ℏ ⎟ k ⃗ ⎥ ρ̃(k ⃗) = Φ0(k) ∑ ∑ dmm H ⃗ ⃗ ′ m⃗ ⎢⎣⎝ 2mkBT ⎠ ⎥⎦ → ⎯ → ⎯ ′ m m 1/2 ⎤ ⎡⎛ ℏ2 ⎞ Hm⃗ ′⎢⎜ ⎟ k ⃗⎥ ⎢⎣⎝ 2mkBT ⎠ ⎥⎦

(9)

where 9610


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The Journal of Physical Chemistry B ⎛ ℏ2 ⎞3/2 ⎛ ℏ2k 2 ⎞ Φ0(k) = ⎜ ⎟ exp⎜ − ⎟⇒ ⎝ 2πmkBT ⎠ ⎝ 2mkBT ⎠

∞

∫−∞

d3k ρ ̃(k ⃗)

⎛ ℏ2 ⎞1/2 1 ⎜ ⎟ {n01( r ⃗ , t )kx n00( r ⃗ , t ) ⎝ mkBT ⎠

f1 ( r ⃗ , k ⃗ , t ) =

(10)

⎛ ℏ2 ⎞1/2 + n02( r ⃗ , t )k y + n03( r ⃗ , t )kz + ⎜ ⎟ ⎝ mkBT ⎠

eq 9 allows us to obtain the following exact series expansion for ⃗ the positive-valued one-particle distribution function, f(r,⃗ k,t).

[n11( r ⃗ , t )kx 2 + n22( r ⃗ , t )k y 2 + n33( r ⃗ , t )kz 2

=

∑ dmm⃗ ⃗ → ⎯ m

f (r ⃗ , k ⃗ , t) =

+ n12( r ⃗ , t )kxk y + n13( r ⃗ , t )kxkz + n23( r ⃗ , t )k ykz]}

∑ n j ⃗ j ⃗ ( r ⃗ , t ) Π j ⃗ j ⃗ (k ⃗ )

⃗ is as given in eq 15. Finally, we note that eq 11 where f(0)(r,⃗ k,t) is, caeteris paribus, the expansion based on tensor Hermite polynomials as elaborated by Grad.52,53,79

j⃗

+

∑ ∑ Re[n j ⃗ l ⃗( r ⃗ , t )]Π j ⃗ l ⃗(k ⃗) j⃗

l ⃗≠ j ⃗

(11)

IV. COLLISION OPERATOR We here define and evaluate the collision operator, Ĝ ab, of eq 1 and then advance a simplification scheme to finally obtain a linear approximation of the Boltzmann transport equation. Let us consider the scattering event in which the initial wave vector k1⃗ /k2⃗ of the molecular species a/b changes to k1′⃗ /k2′⃗ . In the present study, we ignore the effect of quantum statistics on the distributions of molecular species;62 that is, we do not identify a given molecular species either as a boson or a Fermion. The collision-induced change in the distribution function, fa(r,⃗ k1⃗ ,t), for the molecular species a, under the Stosszahlansatz of Boltzmann and the framework of binary collision with molecular species b, is then effected by the collision operator Ĝ ab, as defined below.51

where Π j ⃗ l ⃗ (k ⃗ ) = Φ j ⃗ (k ⃗ )Φ l ⃗ (k ⃗ ) ⇒

∞

∫−∞ d3k Π j ⃗ l ⃗(k ⃗) = δ j ⃗ l ⃗

(12)

⎛ ℏ2 ⎞3/4 ⎛ ℏ2k 2 ⎞ ⎡⎛ ℏ2 ⎞1/2 ⎤ ⃗ Φ j ⃗ (k ) = ⎜ ⎟ exp⎜ − ⎟H j ⃗ ⎢⎜ ⎟ k ⃗⎥ ⎥⎦ ⎝ 2πmkBT ⎠ ⎝ 4mkBT ⎠ ⎢⎣⎝ 2mkBT ⎠ (13)

and ∞

n( r ⃗ , t ) =

∫−∞ d3k f ( r ⃗ , k ⃗ , t ) = ∑ n j ⃗ j ⃗( r ⃗ , t ) j⃗

(14)

̂ f ( r ⃗ , k1⃗ , t ) = Gab a

The double infinite series in eq 11 is an essential feature that ⃗ is positive-valued, we note. The summation ensures that f(r,⃗ k,t) symbol j ⃗ in eq 11 stands for a triplet of integers: 0 ≤ jx < ∞, 0 ≤ jy < ∞, and 0 ≤ jz < ∞. The term with jx = jy = jz = 0 defines ⃗ which is locally the one-particle distribution function, f(0)(r,⃗ k,t), in thermal equilibrium, as shown below. f

(0)

(17)

× [fa ( r ⃗ , k1⃗ ′, t )fb ( r ⃗ , k 2⃗ ′, t ) − fa ( r ⃗ , k1⃗ , t )fb ( r ⃗ , k 2⃗ , t )] (18)

where f b(r,⃗ k2⃗ ,t) is the distribution function for the molecular species b and σab(k1⃗ ,k2⃗ → k′1⃗ k′2⃗ ) is related to the quantum transition operator, T̂ , as follows.

⎛ ℏ2 ⎞3/2 ⎛ ℏ2k 2 ⎞ ⃗ ( r ⃗ , k , t ) = n00( r ⃗ , t )⎜ ⎟ exp⎜ − ⎟ ⎝ 2πmkBT ⎠ ⎝ 2mkBT ⎠

2ℏ δ (k1⃗ + k 2⃗ − k1⃗ ′ − k 2⃗ ′) ⎛ (k )2 − (k ′)2 (k )2 − (k 2′)2 ⎞ 1 ⎟ δ⎜ 1 + 2 ma mb ⎠ ⎝

σab(k1⃗ , k 2⃗ → k1⃗ ′k 2⃗ ′) = (15)

Notably, if the summation index j ⃗ spans the set of N triplets of integers then the total number of independent terms in eq 11 will be N(N + 1)/2. As an explicit example, let the summation indices j ⃗ and l ⃗ span the set, {(0,0,0), (1,0,0), (0,1,0) and (0,0,1)}; that is, N = 4. There will then be a total of 10 terms in the expansion in eq 11, as we show below. For the set under consideration, we have the following values for the Hermite polynomials: H000 = 1, H100 = ℏkx/(mkBT)1/2, H010 = ℏky/ (mkBT)1/2 and H001 = ℏkz/(mkBT)1/2. That means, H0 0 0 H0 0 0 = 1, H0 0 0 H1 0 0 = ℏkx/(mkBT)1/2, H0 0 0 H0 1 0 = ℏky/(mkBT)1/2, H0 0 0 H0 0 1 = ℏkz/(mkBT)1/2, H1 0 0 H1 0 0 = ℏ2kx2/mkBT, H1 0 0 H0 1 0 = ℏ2kxky/mkBT, H1 0 0 H0 0 1 = ℏ2kxkz/mkBT, H0 1 0 H0 1 0 = ℏ2ky2/mkBT, H0 1 0 H0 0 1 = ℏ2kykz/mkBT and H0 0 1 H0 0 1 = ℏ2kz2/mkBT. eq 11 then takes the following expression. f (r ⃗ , k ⃗ , t) = f

(0)

(r ⃗ , k ⃗ , t) + f

(1)

×

(3)

⎛ 2π ⎞2 ⃗ ′ ⃗ ′ ̂ ⃗ ⃗ ⎟ ⟨k , k |T |k , k ⟩ 1 2 ⎝ℏ⎠ 1 2

2

⎜

(19)

where 1 exp[i(k1⃗ · ra⃗ + k 2⃗ · rb⃗ )] (2π )3

⟨ ra⃗ , rb⃗ |k1⃗ , k 2⃗ ⟩ =

(20)

and ∞

T̂ =

(+)

∑ V̂ (Ĝ n=1

V̂ )n − 1 (21)

In eq 21, Ĝ (+) is the causal free particle quantum Green’s function and V̂ is the potential energy operator between the colliding molecules. In the position representation, the potential operator is frequently a function of the intermolecular separation r ⃗ = ra⃗ −rb⃗ and hence it is expedient to define the relative and the total wave vectors as given below.

(r ⃗ , k ⃗ , t)

= f (0) ( r ⃗ , k ⃗ , t )[1 + f1 ( r ⃗ , k ⃗ , t )]

∫ ∫ ∫ d3k 2 d3k1′ d3k 2′ σab(k1⃗ , k 2⃗ → k1⃗ ′k 2⃗ ′)

(16)

where 9611


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The Journal of Physical Chemistry B ma k T⃗ − kR⃗ ma + mb

(22)

δl(kR ) =

mb k 2⃗ = k T⃗ + kR⃗ ma + mb

(23)

αl =

k1⃗ =

With eqs 20, 22, and 23, the expression for σab in eq 19 simplifies to the following form.

x jl (x)̃ x ̃ jl + 1 (x)̃

∑ (−1)k

a 2k (l + 1/2)

k=0

ℏ δ(kR − kR′ ) (3) ⃗ = δ (k T − k T⃗ ′)|Sab(kR⃗ , kR⃗ ′ )|2 μab kR′

[(l − 1)/2]

Bl (x) =

with

ak (l + 1/2) =

⎛ 2π ⎞ Sab(kR⃗ , kR⃗′ ) = −⎜ ⎟ μab ⟨kR⃗ ′|T̂ |kR⃗ ⟩ ⎝ℏ⎠ 2

In eq 25, μab = mamb/(ma + mb) is the reduced mass for the pair of molecules. The quantity Sab(kR⃗ ,kR′⃗ ) in eq 25 is known as the scattering amplitude, the modulus square of which is the differential cross section for the scattering event in question.56−58 Using eqs 22 and 23, it is easy to demonstrate the invariance of the integration measure; that is, d3k1′ d3k2′ = d3kT′ d3kR′ . And therefore, the collision operator in eq 18 may be expressed as follows.

(31)

(l + k )! 1 (k ≤ l ) (l − k)! 2kk! (32)

(33)

⎡ ⎛ 2 ⎤ ⎞ k 1 k δ1(kR ) = arctan⎢ ⎜⎜ R 2 − 1⎟⎟ − R cot(kR̃ s)⎥ − kRs ⎢⎣ kRs ⎝ k ̃ ⎥⎦ ⎠ kR̃ R π + (34) 2 Furthermore, if we assume kR/ kR̃ ≪ 1, then eqs 33 and 34 simplify as follows:

(26)

Next, to obtain an explicit expression for Sab(kR⃗ ,k′R⃗ ) to be inserted in eq 26, we consider the collision between molecules to be elastic in nature. That means, the total linear wave vector kT⃗ (=k1⃗ + k2⃗ ) of the colliding partners and the magnitude of the relative wave vector kR⃗ , which is equal to μ k 2⃗ /mb − k1⃗ /ma , ab

δ0(kR ) ≈ −kRs

(35)

δ1(kR ) ≈ π /2 − kRs − arctan[1/(kRs)]

(36)

We now outline a controlled sequence of well-defined simplification scheme to obtain an analytical expression for the collision integral in eq 26, useful for practical transport studies. As eq 26 reveals, the collision operator, Ĝ ab, depends on the yet unknown distribution function, fa(r,⃗ k1⃗ , t). To remove this nonlinearity, we first note that one is frequently interested to understand the nonequilibrium phenomenon with reference to the corresponding situation in thermal equilibrium, and therefore, we decompose the unknown distribution function, fa(r,⃗ k1⃗ , t), as follows:

are the constants of motion, and as a consequence there is a change, as a result of collision, only in the direction of the relative wave vector, kR⃗ , of the colliding partners. Furthermore, we also assume that the interaction potential V̂ in eq 21 is spherically symmetric. Under these circumstances, there are a number of quantum theoretic approaches to compute the energy-dependent cross sections for elastic collisions.56−58,81 For the sake of simplicity, we here use the well-known partial wave technique, which yields the following expression for the scattering amplitude:56,57

sin[δl(kR )]Pl(cos θ )

x 2k + 2

⎡k ⎤ δ0(kR ) = arctan⎢ R tan(kR̃ s)⎥ − kRs ⎣ kR̃ ⎦

δ(kR − kR′ ) kR′

∞

∑ (2l + 1)eiδ (k ) l

a 2k + 1(l + 1/2)

where x = kR s and x̃= kR̃ s with kR̃ = (1/ℏ) 2μ(E − V0) ; that is, kR2 = kR̃ 2 + 2μV0/ℏ2. jl(x) in eq 29 is the spherical Bessel function of the first kind. Special values for the phase shift are as given below.

|Sab(kR⃗ , kR⃗ ′ )|2 × [fa ( r ⃗ , k1⃗ ′, t )fb ( r ⃗ , k 2⃗ ′, t ) − fa ( r ⃗ , k1⃗ , t )

1 Sab(kR⃗ , kR⃗ ′ ) = Sab(kR , θ ) = kR

(30)

= 0(k > l)

(25)

fb ( r ⃗ , k 2⃗ , t )]

( −1)k

x 2k + 1

k=0

(24)

∫ ∫ d3k 2d3kR′

∑

(28)

(29)

[l /2]

A l (x ) =

σab(k1⃗ , k 2⃗ → k1⃗ ′k 2⃗ ′)

̂ f ( r ⃗ , k1⃗ , t ) = ℏ Gab a μab

⎛ B (x) + αlAl + 1(x) ⎞ lπ − x − arctan⎜ l ⎟ 2 ⎝ Al (x) − αlBl + 1(x) ⎠

fa ( r ⃗ , k1⃗ , t ) = f a(0) ( r ⃗ , k1⃗ , 0) + f a(1) ( r ⃗ , k1⃗ , t )

(37)

(a) f a(0) ( r ⃗ , k1⃗ , 0) = n00 ( r ⃗ , 0)ϕ0(a)(k1⃗ )

(38)

⎛ ℏ2k 2 ⎞ ⎛ ℏ2 ⎞3/2 1 ⎟ ϕ0(a)(k1⃗ ) = ⎜ ⎟ exp⎜ − ⎝ 2πmakBT ⎠ ⎝ 2makBT ⎠

(39)

R

l=0

(27)

Here θ is the angle between the relative wave vectors before and after the collision. The effect of interaction potential is then fully contained in the quantity, δl(kR)frequently known as the phase shiftwhich may be obtained either analytically or by numerical means. Here, for simplicity, we assume the intermolecular potential to be of finite range; that is, V(r) = 0 for r > s and V(r) = V0 for r ≤ s, in which case the analytical expression for the phase shift is as given below.57

where (r,⃗ k1⃗ ,t) represents the time-dependent fluctuation in the distribution function from its initial local equilibrium value. The decomposition in eq 37 leaves the possibility that the system, at time t = 0, need not necessarily be in local thermal equilibrium. Next, the dynamics of molecular species in a binary mixture, according to the Boltzmann equation, are coupled f(1) a

9612


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The Journal of Physical Chemistry B through the collision integrals. To decouple the equation of motion for one molecular species, therefore, we assume the fluctuations in the distribution function, from its initial local equilibrium value, for the other species to be vanishingly small; that is, f b(r,⃗ k,⃗ t) ≈ f b(0)(r,⃗ k,⃗ 0). This would be the case, for example, when the molecular species b, in a given situation, constitute a large bath, within which the dynamics of molecular species a happens. We now use eq 37 and the constraint imposed by the law of energy conservation (k12/ ma + k22/mb = k1′2/ma + k2′2/mb) to obtain the following identity for the integrand of eq 26.

∇⃗kR⃗ ϕ0(b)(k 2⃗ ) =

(45)

ϕ0(b)(k 2⃗ ′) − ϕ0(b)(k 2⃗ ) ≈ (kR⃗ ′ − kR⃗ )·[∇⃗kR⃗ ϕ0(b)(k 2⃗ )]kR⃗ = kR⃗′

0

(46) (b) To reach eq 46, we have used the expressions for ϕ(a) 0 and ϕ0 (av) as given in eq 39. Let us now define the average, Gab , of the frequency, Wab, in eq 44 as follows (see Supporting Information, section II).

(av) Gab =

∫ d3k1 Wabϕ0(a)(k1⃗ )

= − nb( r ⃗ , 0)

With eq 41, eq 40 transforms to the following form.

4π ℏ3 μab 2 kBT

⎡ sin 2 δ0(kR )⎢1 + ⎢⎣

fa ( r ⃗ , k1⃗ ′, t )fb ( r ⃗ , k 2⃗ ′, t ) − fa ( r ⃗ , k1⃗ , t )fb ( r ⃗ , k 2⃗ , t ) = {[f b(0) ( r ⃗ , k 2⃗ ′, 0) − f b(0) ( r ⃗ , k 2⃗ , 0)] + f b(0) ( r ⃗ , k 2⃗ ′, 0) (kR⃗′ − kR⃗ ) ·∇⃗kR⃗ [ln f a(1) ( r ⃗ , k1⃗ ′, t )]} × f a(1) ( r ⃗ , k1⃗ , t )

∞

−

(42)

∞

⎛ sin δl ⎞2 ⎟ ⎝ sin δ0 ⎠

∑ (2l + 1)⎜ l=1

⎤ ⎛ sin δl ⎞⎛ sin δl + 1 ⎞ ⎟⎜ ⎟ cos(δl − δl + 1)⎥ ⎥⎦ ⎝ sin δ0 ⎠⎝ sin δ0 ⎠

To go from eq 47 to 48, we have substituted eq 27 for the scattering amplitude in eq 44. Now, from eqs 35 and 36, it is easy to show that the leading term of the ratio, sin δ1(x)/ sin δ0(x), for small x, is x2/2 and therefore eq 48 may be further approximated as follows.

(43)

(av) Gab ≈ −nb( r ⃗ , 0)

× [ϕ0(b)(k 2⃗ ′) − ϕ0(b)(k 2⃗ )]

∫ ∫ d3k1 d3k 2 kRϕ0(a)(k1⃗ )ϕ0(b)(k 2⃗ )

(48)

with

∫ ∫ d3k 2 d3kR′

(47)

∑ 2(l + 1)⎜ l=0

Evidently, the nonlinearity in the collision operator, in eq 26, arises from the logarithmic term in eq 42. Now, the logarithm is a slowly varying function and hence the term involving the gradient in eq 42 may safely be dropped. With the set of approximations outlined above, the collision operator in eq 26 simplifies to the following form. ̂ f ( r ⃗ , k1⃗ , t ) ≈ Wabf (1) ( r ⃗ , k1⃗ , t ) Gab a a

ℏ2 ⃗ ′ ⃗ kR ·(kR − kR⃗ ′ )ϕ0(b)(k 2⃗ ) μab kBT

=

≈ 1 + (kR⃗ ′ − kR⃗ ) ·∇⃗kR⃗ ln f a(1) ( r ⃗ , k1⃗ ′, t )

ℏnb( r ⃗ , 0) μ

(kR⃗ ′ − kR⃗ ) ·(∇⃗kR⃗ [ϕ0(a)(k1⃗ )ϕ0(b)(k 2⃗ )])kR⃗ = kR⃗′ ϕ (a)(k1⃗ )

≈

(40)

(41)

Wab =

∇⃗kR⃗ [ϕ0(a)(k1⃗ )

In eq 45, the second term on the right-hand side has been ignored because the logarithm is a slowly varying function. We (b) ⃗ ⃗ now expand ϕ(b) ′⃗ and use eq 45 and 0 (k2) ≡ϕ0 (kR) around kR the conservation of the total linear wave vector (kT⃗ = kT′⃗ ) to obtain the following approximate identity.

Now, the integrand in eq 26 also includes the scattering cross section |Sab (kR, θ) |2, which is usually small for the large scattering angle θ (the angle between kR⃗ and k′R⃗ ). Therefore, as noted by Bethe, Rose, and Smith,84 the most dominant contribution to the collision integral is expected to come from those k′R⃗ which are close to kR⃗ ; that is k′R⃗ − kR⃗ = ω⃗ with ω (=2 sin θ/2) ≪ 1. We may then make a Taylor expansion of ⃗ f(1) a (r,⃗ k1,t) in powers of ω⃗ (always on the surface of the unit sphere) and obtain the following simplifying approximation.

f a(1) ( r ⃗ , k1⃗ , t )

1 ϕ0(a)(k1⃗ )

ϕ0(b)(k 2⃗ )]

⎡ f (1) ( r ⃗ , k ⃗ ′, t ) ⎤ 1 = ⎢ a(1) f b(0) ( r ⃗ , k 2⃗ ′, 0) − f b(0) ( r ⃗ , k 2⃗ , 0)⎥ ⎢⎣ f ( r ⃗ , k ⃗ , t ) ⎥⎦ 1 a

f a(1) ( r ⃗ , k1⃗ ′, t )

∇⃗kR⃗ [ϕ0(a)(k1⃗ )ϕ0(b)(k 2⃗ )]

− ϕ0(b)(k 2⃗ )∇⃗kR⃗ [ln ϕ0(a)(k1⃗ )] ≈

fa ( r ⃗ , k1⃗ ′, t )fb ( r ⃗ , k 2⃗ ′, t ) − fa ( r ⃗ , k1⃗ , t )fb ( r ⃗ , k 2⃗ , t )

f a(1) ( r ⃗ , k1⃗ , t )

1 ϕ0(a)(k1⃗ )

δ(kR − kR′ ) |Sab(kR , θ )|2 kR′

sin 2 δ0(kR )

4π ℏ3 μab2 kBT

∫ ∫ d3k1 d3k 2 kRϕ0(a)(k1⃗ )ϕ0(b)(k 2⃗ ) (49)

(b) ⃗ ⃗ We now use the expression for ϕ(a) 0 (k1) and ϕ0 (k1) as given in eq 39 and perform the integration in eq 49 analytically to obtain the following expression for the average of the collision operator.

(44)

where Wab stands for the generalized frequency, associated with the collision between the molecule a and molecule b. To simplify eq 44 further, let us consider the following identity. 9613


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The Journal of Physical Chemistry B (av) Gab

⎛∂ ⎞ ℏ ⃗ ⃗ (av) (av) (1) − Gab k1·∇r ⃗ − Gaa ⎜ + ⎟f a ( r ⃗ , k1⃗ , t ) ma ⎝ ∂t ⎠

⎛ 2π ℏ2 ⎞1/2 ⎟⎟ = −nb( r ⃗ , 0)⎜⎜ ⎝ μab kBT ⎠

⎛ 2μ kBT ⎞ ⎛ 3 1 2μ kBT ⎞⎤ 4ℏ ⎡ ⎢1 − exp⎜ − ab 2 s 2⎟1F1⎜ − ; ; ab 2 s 2⎟⎥ × μab ⎣ ⎝ ⎠ ⎝ 2 2 ⎠⎦ ℏ ℏ =

−Nb(th)( r ⃗ ,

+

(50)

where the thermal frequency, ωth = (2/π)kBT/ℏ. 0) = (λth) 3 nb(r,⃗ 0) is the number of molecular species b within the thermal volume (λth) 3 where the thermal length, λth= [2πℏ2/ (μab kBT)]1/2. The dimensionless quantity ϵ = 4π(s/λth) 2 is the ratio of the total scattering cross section and the thermal area. For simplicity, we may also choose nb(r,⃗ 0) = n(av) b , which is the spatially averaged density of the molecular species b. Notably, n(av) is equal to [ñb(g,⃗ 0)]g⃗=0, where ñb(g,⃗ 0) is the spatial Fourier b transform of nb(r,⃗ 0). The symbol 1F1 in eq 50 stands for the confluent hypergeometric function. If we now retain only the leading terms involving ϵ, eq 50 further simplifies as follows. N(th) b (r,⃗

iℏ f a(1) (g ⃗ , k1⃗ , t ) = e Ωt f a(1) (g ⃗ , k1⃗ , 0) − (k1⃗ ·g ⃗)(e Ωt − 1) ma Ω f a(0) (g ⃗ , k1⃗ , 0)

(av) Gab

b

(eff) (1) ̂ f ( r ⃗ , k1⃗ , t ) ≈ −ωscat f a ( r ⃗ , k1⃗ , t ) ⇒∑ Gab a b

(58)

∂ (1) f (g ⃗ , k1⃗ , t ) ∂t a ⎡ ⎤ iℏ ⃗ (k1·g ⃗)f a(0) (g ⃗ , k1⃗ , 0)⎥ = e Ωt ⎢Ωf a(1) (g ⃗ , k1⃗ , 0) − ma ⎣ ⎦

(59)

⇒

(51)

⎛ iℏt ⎞⎡ iℏ ⃗ =exp(At ) exp⎜ − k1⃗ ·g ⃗⎟⎢Af a(1) (g ⃗ , k1⃗ , 0) − (k1·g ⃗) ma ⎝ ma ⎠⎣ ⎤ fa (g ⃗ , k1⃗ , 0)⎥ ⎦ (60) (1) ⃗ ⃗ We now use eqs 15-17 for f(0) a (g,⃗ k1, 0) and fa (g,⃗ k1, 0) in eq 60 and perform the integration over k1⃗ fully analytically. The resulting expression along with eq 2 and the Fourier relation, nij(−g)⃗ = nij*(g)⃗ , finally yields the following expression for the slope of the MSD (see Supporting Information, section III).

(52)

μ(t ) ≈

(53)

∞

4Lx 2

(eff) (eff) ωscat exp( −ωscat t) na(av)

∑ y(gx)F(gx , t ) gx = 0

(eff) 2 2 exp[− (1/2)(ωscat t ) η (gx )]

Finally, eq 43 may be written in a compact notation as follows. (av) (1) ̂ f ( r ⃗ , k1⃗ , t ) ≈ Gab Gab f a ( r ⃗ , k1⃗ , t ) a

iℏ ⃗ iℏ ⃗ (k1·g ⃗) = A − (k1·g ⃗) ma ma

(av) (av) Ω = Gaa + Gab −

For a number of distinct scattering events taking place in the system, we may define an effective frequency as follows. (eff) (av) ωscat = −∑ Gab

(57)

where

Notably, eq 51 would also result if we consider the molecular scattering events to take place at extremely low energies, in which case the quantity s = (ϵ/π)1/2λth/2 would stand for the (ab) scattering length, lscat , which, if so required, may be independently determined from the scattering experiments. This distinction is important for l(ab) scat and the range of the potential s may, in principle, differ by orders of magnitude.57,85,86 eq 51 may then also be written as follows. ⎛ 2πk T ⎞1/2 (ab) 2 B ⎟⎟ (lscat = −32nb(av)⎜⎜ ) μ ⎝ ab ⎠

(56)

(av) where G(av) aa and Gab are the average of the collision operator as ⃗ given in eq 50, for example. In the Fourier space, f(1) a (r,⃗ k,t) = ∑g⃗ ⃗ f(1) (g , k , t) exp[i(g · r ) ] and therefore the solution of eq 56 is ⃗ ⃗ ⃗ a 1 obtained as follows.

0)ωth[1 − exp( −ϵ)1F1( −3/2; 1/2; ϵ)]

(av) Gab = − 4Nb(th)ωth ϵ = −16πNb(th)ωth(s /λth)2

ℏ ⃗ ⃗ (0) k1·∇r ⃗ f a ( r ⃗ , k1⃗ , 0) = 0 ma

(61)

where

(54)

cos(gx Lx) 1 and y(gx ≠ 0) = 12 gx 2Lx 2

y(gx = 0) = (55)

(62)

3

F (g x , t ) =

where G(av) ab is as given in eq 50 or eq 52.

(eff) m ∑ αm(gx)[η(gx)ωscat t ] with F(gx = 0, t ) m=0

= α0(gx = 0)

V. SLOPE AND CURVATURE We here use eq 54 to define and obtain a complete solution of the linear Boltzmann equation and then determine an analytical expression for the slope, μ(t), of the mean square displacement (MSD) as given in eq 2. Let us consider a binary mixture consisting of two distinct molecular species, denoted here as a and b. Following eq 37, we decompose the distribution function (1) ⃗ ⃗ ⃗ fa(r,⃗ k1⃗ ,t) as f(0) a (r,⃗ k1,0) + fa (r,⃗ k1,t), which means ∂fa(r,⃗ k1,t)/∂t = (1) ∂fa (r,⃗ k1⃗ ,t)/∂t. From eqs 1 and 54, the linearized Boltzmann ⃗ equation for the fluctuation term, f(1) a (r,⃗ k1,t), is then given as follows.

(63) 3

(a) α0(gx ) = η(gx )Im[n01 (gx )] − Re ∑ n(jja)(gx ) j=1

(64)

(a) (a) (a) α1(gx ) = −Im[n01 (gx )] − η(gx )Re[n00 (gx ) − n11 (g x ) (a) (a) + n22 (gx ) + n33 (gx )]

(65)

(a) (a) α2(gx ) = Re[n11 (gx )] − η(gx )Im[n01 (gx )]

9614

(66)


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The Journal of Physical Chemistry B (a) α3(gx ) = η(gx )Re[n11 (gx )]

(67)

ηc (gx ) = −

In eqs 61−67, the quantity η(gx) = −(kBT/ma)1/2 gx/ω(eff) scat is a (eff) = dimensionless number. The effective frequency, ωscat (av) −(G(av) +G ), from eq 52, is a positive number. The density aa ab (a) n(a) ij (gx) stands for nij (gx, gy = 0, gz = 0,t = 0). To facilitate further analysis, we re-express the slope, μ(t), from eq 61, as follows. μ(t ) ≈

∞ (eff) ω scat na(av) gx = 0

4Lx2

ψ (t ) =

(eff) m μm (gx )(ωscat t) ]

m=0

(68)

where μm(gx) is a polynomial in η(gx), the first few of which are as given below. 3 (a) μ0 (gx ) = y(gx ){Im[n01 (gx )]η(gx ) − Re ∑ n(jja)(gx )} j=1

(69)

μ1(gx ) = y(gx ){[η(gx ) − η+(gx )][η(gx ) − η−(gx )]}

(70)

with η±(gx ) =

(a) (gx )] −Im[n01

Re[n(̃ a)(gx )]

⎡ 3 (a) ⎛ ⎞1/2 ⎤ Re[n(̃ a)(gx )]Re[∑ j = 1 n(a) ⎢ ⎥ jj njj (gx )] ⎟ ⎜ ⎢1 ± ⎜1 + ⎥ (a) 2 ⎟ (Im[n01 (gx )]) ⎢⎣ ⎝ ⎠ ⎥⎦ (71) 2

μ2 (gx ) = y(gx ){a0 + a1η(gx ) + a 2[η(gx )] + a3[η(gx )]3 } (72)

where (a) (a) (a) (a) n(a) (gx ) − n11 (gx ) + n22 (gx ) + n33 (g x ) ̃ (gx ) = n00

(73)

3

a0 = −Re[∑ n(a) jj (gx )/2] j=1 (a) a1 = Im[3n01 (gx )/2]

(74) (75)

(a) (a) (a) a 2 = Re[n00 (gx ) − n11 (gx )/2 + 3n22 (gx )/2 (a) + 3n33 (gx )/2]

(76)

and (a) a3 = −Im[3n01 (gx )/2]

(77)

On ignoring terms which are quadratic and higher order in (a) (a) time, t, and the circumstances when n(a) 11 (gx) = n22 (gx) = n33 (gx) = 0, the slope from eq 68 and the curvature, ψ, assume the following simple expressions: μ(t ) ≈

4Lx2

(eff) ωscat na(av)

∞

(78)

where (a) μ0 (gx ) = y(gx )Im[n01 (gx )]η(gx )

(79)

μ1(gx ) = y(gx )η(gx )[η(gx ) − ηc (gx )]

(80)

4Lx2 (eff) 2 d μ(t ) ≈ (av) (ωscat ) dt na

(81) ∞

∑ μ1(gx) gx = 0

(82)

VI. CHARACTERISTICS OF DIFFUSION We now discuss the circumstances under which the system would exhibit various diffusive transport processes. For the existence of normal diffusion, it is necessary for the slope to contain a term that is independent of time. Suppose the initial state (at time t = 0) of the system is a local thermal equilibrium (a) (a) ⃗ one; that is, fa(g,⃗ k1⃗ ,0) = f(0) a (g,⃗ k1,0) and therefore nij = nij δi0δj0, in which case μ0(gx) = 0, from eq 69. In this circumstance, as eq 68 reveals, μ(t) would not contain a term, which is independent of time. This means, there will not be a normal diffusive

(eff) t] ∑ [μ0 (gx) + μ1(gx)ωscat gx = 0

(a) Re[n00 (gx )]

As evident from eq 81, the dimensionless quantity ηc(gx) essentially measures the deviation of the initial molecular distribution function from the local thermal equilibrium one. Notably, eq 78 will be a good approximation if ω(eff) scat t is much less than unity, which is always the case if the temperature is (eff) sufficiently low. Also, ωscat , at a given temperature, is proportional to the square of the scattering length, such as l(ab) scat , as eq 52 reveals. Thus, the scattering length is a key physical attribute that, inter alia, determines the time scale, for which eqs 78−82 are valid. We now briefly point out the approximations underlying the above derivation of observables and outline the way one may improve upon the present results, such as given in eq 61. First, while we have introduced a quantum mechanical treatment for the scattering cross sections to obtain an analytical expression for the collision operator, the streaming terms of the Boltzmann equation in eq 1 remain classical in nature. This is due to the ⃗ in eq 1 stands for the fact that the distribution function, fa(r,⃗ k,t), ⃗ probability density in the (r,⃗ k) space, the range of which spans the set of only positive numbers, as opposed to the probability amplitudes (wave functions, in a given representation) of quantum mechanics, the range of which, in general, spans the full complex plane. Accordingly, eq 1, as it stands, may not fully account for the effects, if any, arising due to quantum interferences, which may be of importance in specific physical situations, particularly at low temperatures. For a more realistic treatment of ensemble dynamics, one may use other quantum mechanical Boltzmann-like equations of motion.61 In any event, the present treatment of the Boltzmann transport equation may be viewed as a mixed quantum-classical approach to the ensemble molecular dynamics. Next, we have here considered only elastic collision processes, which completely ignore the internal structure, if any, of the colliding molecules. Inelastic collisions may be important in specific circumstances, in which case one may consider using, for example, the dynamical equations due to Waldman and Snider,87−90 which are valid for dilute polyatomic gases. Furthermore, the partial wave expansion for quantum cross sections used here assumes the intermolecular potential to be spherically symmetric. For nonspherical situations, it is conceptually straightforward to expand, following the works of Köhler and others,88−90 the intermolecular potential in terms of spherical harmonics and obtain a more elaborate expression for the cross sections, in place of eq 27 utilized in the present study.

∞

∑ [∑

(a) 2Im[n01 (gx )]

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The Journal of Physical Chemistry B transport if the system, at t = 0, is locally in a state of thermal equilibrium. If, on the other hand, the initial state is not in thermal equilibrium then ∑∞ gx=0μ0(gx) = ψ0, from eq 69, is

μ(t ) ≈

expected to be nonzero. In fact, ψ0 must eventually yield a positive value for a physically meaningful scenario. To gain further insight into the nature of diffusive transport and to extract an analytical expression for the coefficient of normal diffusion, D, it is sufficiently illuminating to consider a minimal (a) (a) (a) nonequilibrium situation wherein n(a) 01 ≠ 0 and n11 = n22 = n33 = 0. In such an instance, the expression for the slope and the curvature are as given in eqs 78-82, assuming the scattering length for the given collisional event is not too large. Now, from eq 80, ∑∞ gx=0μ1(gx) = ψ yields the curvature signature in eq

=−

(a) 2Im[n01 (gx )] (a) Re[n00 (gx )]

(a) ⎛ kBT ⎞1/2 (av) ⟨n00 ⎛ kBT ⎞1/2 (av) ⟩ = 2⎜ ⎟ lscat (av) = 2⎜ ⎟ lscat ⎝ ma ⎠ ⎝ ma ⎠ na

2Lx2

(eff) ωscat na(av)

⎛ kBT ⎞1/2 (av) D≈⎜ ⎟ lscat ⎝ ma ⎠

(av) lscat

(av) lscat

Dself ≈

(a) 1 pth 1 (av) ( aa) 2 2π ρa (8lscat )

(89)

where ρ(av) is the average mass density of the molecular species a a, p(a)th = (2makBT)1/2 the thermal momentum and l(aa) scat is the scattering length for the collisional event between the molecular species a. For a dilute mixture, on the other hand, we have n(av) a ≪ n(av) and therefore eqs 87 and 88 take the following simple b form for the coefficient of mutual diffusion 1/2 mb ⎛ μab kBT ⎞ 1 1 D≈ ⎜ ⎟ (av) ( ab) 2 ma ⎝ 8π ⎠ ρb (4lscat ) (ab) 1 mb pth 1 = (av) ( ab) 2 π ma ρb (8lscat )

(84)

(90)

ρ(av) b

as follows.

where is the average mass density of the molecular species b, p(ab) = (2 μabkBT)1/2 the thermal momentum and l(ab) th scat is the scattering length for the collisional event between the molecule a and the molecule b, as discussed in section IV. eq 90, in fact, stands for the single molecule diffusion coefficient. If we measure the mass in atomic mass unit (amu), the mass density in kg-meter−3, length in meter and the temperature in Kelvin then eq 90 simplifies as follows.

⎛ kBT ⎞1/2 1 (av) =⎜ ⇒ η(gx ) = −lscat gx and η(gx = 0) ⎟ (eff) ⎝ ma ⎠ ωscat

=0

⎤−1 (av) ⎞ ⎛ (av) ⎞ ⎢⎜ nb ⎟(l (ab))2 + ⎜ na ⎟(l (aa))2 ⎥ ⎜ μ ⎟ scat ⎥ ⎢⎣⎜⎝ μab ⎟⎠ scat ⎝ aa ⎠ ⎦

Notably, eqs 87 and 88 are, mutatis mutandis, fully consistent with the expression for the diffusion coefficient developed independently by Chapman51 and Enskog,54 before the advent of quantum mechanics. Now, in the instance when there is only one molecular species present in the system, eqs 87 and 88 reduce to the coefficient of self-diffusion as given below.

∞

Next, we define the average scattering length,

1/2 ⎡⎛

1 ⎛ 1 ⎞ = ⎜ ⎟ 32 ⎝ 2πma ⎠

(88)

∑ y(gx)Re[n00(a)(gx)]η2(gx) l(av) scat ,

(87)

A simple expression for the average scattering length, l(av) scat , in eq 87 may be obtained from eqs 52 and 85, as given below.

(83)

gx = 0

(86)

(av) (a) where ⟨n(a) 00 ⟩ is the spatially averaged value, na , of n00 (r)⃗ . To reach eq 86, we have taken the density to be an even function. Now, for normal diffusion, the slope must be equal to 2D and therefore we extract the coefficient of diffusion, D, from eq 86 as follows.

With μ1(gx) = 0 and eq 83, eq 78 simplifies as follows. μ(t ) ≈ −

1/2 ∞ 2 ⎛ kBT ⎞ (av) (a) ∑ l cos(gx Lx )Re[n00 (gx )] ⎜ ⎟ scat na(av) ⎝ ma ⎠ g >0 x

(a) ⇒ Im[n01 (gx )]

1 (a) = − η(gx )Re[n00 (gx )] 2

∑ y(gx)Re[n00(a)(gx)]η2(gx)

1/2 2 ⎛ kBT ⎞ (av) (a) = (av) ⎜ (gx = 0)] ⎟ lscat Re[n00 na ⎝ ma ⎠

82, which may be either negative, zero or positive. Accordingly, the system will exhibit a subdiffusion (superdiffusion) phenomenon when ψ0 > 0 and ψ < 0 (ψ > 0). The situation when ψ0 > 0 and ψ = 0 will give rise to normal diffusion and this will necessarily happen when η(gx) = ηc(gx), as eq 80 reveals. Clearly, then, η(gx) (and therefore, from eqs 52 and 53, the scattering length) is the physical tuning parameter and ηc(gx), which measures the deviation of the molecular state (at time t = 0) from the local thermal equilibrium one as shown in eq 81, defines the critical point that separates the normal and anomalous diffusion regimes. Analogous to continuous phase transitions in magnetic materials,59,60 we may then view the curvature, ψ(t), as an appropriate order parameter to monitor the diffusive transport processes in nature. The present elaboration on anomalous diffusion complements the stochastic approach to diffusion, wherein the mean square displacement is frequently postulated to have the form, ⟨x2⟩∝ tα with α > 1 (positive curvature) and α < 1 (negative curvature) respectively standing for superdiffusion and subdiffusion.7−9,20,83 To extract the coefficient of diffusion, D, let us consider the critical point η(gx) = ηc(gx) and therefore μ1(gx) = 0, in which case we have the following relation, from eq 81. η(gx ) = ηc (gx ) = −

∞ (eff) − (av) ωscat na gx > 0

2Lx2

(85)

With the definition in eq 85, eq 84 simplifies as follows. 9616


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The Journal of Physical Chemistry B D≈

imation of the quantum elastic scattering amplitude (with k⃗ ≈ ⃗ as given below.56−58 k′), ∞ ∞ μ μ ⃗ ⃗ (ab) ≈ ab 2 lscat d3r ei(k − k ′)· r ⃗v( r ⃗) ≈ ab 2 d3r v( r ⃗) −∞ −∞ 2π ℏ 2π ℏ

⎤ 1 T1/2 ⎡ mb 1/2 μ ( ) ⎢ ⎥ (ab) 2 ab ⎦ (lscat ) ρb(av) ⎣ ma

× 1.88767048 × 10−27 m 2 s−1

∫

(91)

(94)

Furthermore, if we use the Stokes−Einstein relation,1,3,5 the friction coefficient, γ, may be obtained from eq 90 as follows: ma ⎛ 8πkBT ⎞ ⎜⎜ ⎟⎟ mb ⎝ μab ⎠

Here v(r)⃗ stands for the potential energy between the molecular species a and b. We now substitute eq 94 in eq 90 to obtain the following expression for the average potential energy.

1/2

γ = ρb(av)

⇒ηv =

(ab) 2 (4lscat )

1/2 (ab) 2 1 (av) ma ⎛ 2kBT ⎞ (4lscat ) ⎜⎜ ⎟⎟ ρb 3 mb ⎝ πμab ⎠ RH

∫

(92)

1/4 ⎛ ⎞1/2 ℏ2 ⎛ πkBT ⎞ ⎜ (ma + mb)π ⎟ ⎜⎜ ⎟⎟ d r v( r ⃗ ) ≈ 2ma ⎝ 2μab ⎠ ⎜⎝ 2ρb(av)D ⎟⎠ −∞

∫ (93)

∞

3

(95)

1/4 ⎛ ⎞1/2 1 ⎛ T ⎞ ⎜ ma + mb ⎟ = ⎜⎜ ⎟⎟ ⎜ (av) ⎟ × 1.82829499 × 10−54 J m 3 ma ⎝ μab ⎠ ⎝ ρb D ⎠

Here ηv is the viscosity of the medium within which the diffusion occurs and RH is the Stokes (hydrodynamic) radius of the diffusing molecular species. To examine the numerical accuracy of eq 90, let us consider, as an example, the diffusion of potassium ion (atomic mass = 39.0983 amu) in water (molecular mass = 18.0153 amu and density = 996.5162 kg/m3), a dynamical phenomenon of interest in biological cell membrane. We take l(ab) scat to be the range, s, of the potential (see eq 51), which we estimate to be somewhere between 1.60 × 10−10 and 1.80 × 10−10 m (an approximate location of the hard repulsive potential wall), from the results of detailed ab initio electronic structure calculations.91,92 At T = 300 K, eq 90 then yields the diffusion coefficient, D = (1.64−2.07) × 10−9 m2/s, which is close to the experimental value45 of about 2 × 10−9 m2/s. Clearly, the numerical value of D, inter alia, critically depends on the scattering length, l(ab) scat , which, in turn, is a function of the microscopic dynamics of the diffusing molecular species. As we have pointed out earlier, l(ab) scat for different molecular systems may vary considerably and it may differ from the range, s, of the microscopic interaction potential by orders of magnitude.57 This is frequently the case, for example, at ultracold temperatures in the Bose−Einstein condensates.85,86 Thus, the scattering length may faithfully be used as a fundamental physical attribute to rationalize wide numerical variations in the diffusion coefficient, often encountered in biological systems. For example, the experimental D for the green fluorescent protein (GFP) in the cytoplasm of E. coli46 is about 7 × 10−12 −9 m2/s, implying the l(ab) scat for GFP in cytoplasm to be 2.75 × 10 m at T = 300 K. As yet another example, the experimental D for the interphase chromatin in the living cells of S. cerevisiae47 is about 5 × 10−16 m2/s, which indicates the l(ab) scat for chromatin in the yeast nucleus to be about 3.26 × 10−7 m at T = 300 K. Experimentally derived scattering length, thus, yields a new perspective to infer the, often unknown, effective range of interactions among the constituent molecular species in a complex system. In any event, eqs 87−90 provide a simple, yet nontrivial, correlation between the microscopic (quantum dynamics) and macroscopic (diffusive transport) molecular world. To gain further insight into the diffusive transport processes, we now ask: what is the magnitude of the effective potential energy, if any, that the molecular species a experiences during the diffusional dynamics taking place at a given temperature? To obtain a simple, yet nontrivial, answer to this question, we here solve a preliminary inverse scattering problem.93 As a first approximation, for example, we may fruitfully associate the scattering length, l(ab) scat , with the standard first Born approx-

(96)

If we assume the interaction potential to be spherically symmetric, of the form, e.g., v(r)⃗ = v0 exp(−r/r0) then eqs 95 yields the expression for v0 as shown below. 1/4 ⎛ ⎞1/2 1 ℏ2 ⎛ πkBT ⎞ ⎜ ma + mb ⎟ ⎟⎟ v0 = 3 ⎜⎜ 8r0 2ma ⎝ 2μab ⎠ ⎜⎝ 2πρb(av)D ⎟⎠

(97)

1/4 ⎛ ⎞1/2 1 ⎛ T ⎞ ⎜ ma + mb ⎟ = 3 ⎜⎜ ⎟⎟ ⎜ (av) ⎟ × 7.27455252 × 10−56 J mar0 ⎝ μab ⎠ ⎝ ρb D ⎠

(98)

In eqs 96 and 98, the mass is expressed in atomic mass unit (amu), mass density in kilogram per cubic meter, length in meters, and the temperature in kelvin. We may interpret v0 in eq 97 as the temperature-dependent effective potential energy which the molecular species a experiences during the course of diffusion. And therefore, v0, as given, for example, in eq 97, may be considered as the dynamical cause of the perpetual molecular movement, which finally emerges as the diffusive transport phenomena observed in nature. This is a valuable information for it is extremely difficult to obtain an estimate of the interacting energy, acting among the complex molecular species in a nontrivial environmental setting, by other theoretical means. To estimate a representative value for v0, we continue with the example of the diffusion of potassium ion in water at T = 300 K. With the experimental value for the diffusion constant (D = 2 × 10−9 m2/s)45 and r0 = 5 × 10−10 m as a conservative estimate from the results of detailed ab initio electronic structure calculations,91,92 eq 96 yields, v0 = 1.76957 × 10−25 J, which corresponds to a voltage difference, NAv0/F = 1.1 × 10−3 mV (NA = Avogadro number and F = 1 farad). This is, as expected for pure diffusion, much smaller than RT/F (≈25.85 mV at 300 K), which gives the order of magnitude frequently encountered for the drift, superimposed over the pure diffusion processes in biological ion channels.94 Accordingly, the phenomenon of molecular diffusion may be viewed as a (temperature-dependent) random motion, wherein the ensemble of diffusing molecules move under the influence of an effective potential energy (temperature-dependent diffusion potential), of the order of 0.1 J/mol (equivalently, 1 μV) or less in a biological environment. To summarize, a tentative physical picture of normal diffusion that emerges from the above analysis is as follows. 9617


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The Journal of Physical Chemistry B Because of ever-present thermal agitations and numerous collisions with surrounding molecules, the ensemble of molecular species a experiences an effective, relatively much smaller in magnitude, temperature-dependent potential energy, which is distributed around v0, as given, for example, in eq 97. This distribution of effective interaction energy, inter alia, may be considered as a cause underlying the perpetual and random motion of individual molecular species a, within the ensemble. As a consequence of molecular movements, due to the distribution around v0, the local concentration (equivalently, the local molecular density) of species a fluctuates in time, leading to a dynamical concentration gradient in the system. This space-time fluctuation of the molecular density then eventually gives the appearance of an ordered motion of molecules down the dynamic concentration gradient (à la Fick44), finally leading to what we observe as the diffusion. The idea of the effective temperature-dependent potential energy (equivalently, the effective voltage) outlined above may be used to build an effective dynamical model (a Hamiltonian, for example) to describe the phenomenon of diffusive molecular dynamics in nature, we surmise.

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AUTHOR INFORMATION

Corresponding Author

*(A.V.) E-mail: [email protected]. Telephone: +91 44 22574234. Fax: 91 44 22574202. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS S.M. acknowledges Indian Institute of Technology Madras, India for fellowship. S.C. acknowledges financial support from the Council of Scientific and Industrial Research, New Delhi, India, under Grant No. 09/084(0513)/2009-EMR-I.

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REFERENCES

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VII. CONCLUSION We now briefly discuss how the present elaboration of the Boltzmann transport theory, in conjunction with quantum scattering, may be further extended and utilized to study the diffusive transport processes, particularly in biological systems. For simplicity, we have here evaluated the collision integral using the partial wave expression for the quantum scattering amplitude and obtained a linear form of the transport equation to study diffusion. This is a reasonable good approximation as the present study has revealed. The expression for the coefficient of diffusion in terms of quantum scattering length as given, for example, in eq 90 is of immediate use to rationalize diffusional transport studies in a variety of context, including the biological systems. The notion of scattering length appears as a fundamental dynamical attribute of the system and it provides a legitimate and useful conceptual framework, which may be faithfully used to understand wide numerical variations in the coefficient of diffusion, frequently encountered within the crowded environment of biological cells. The expression for the collision operator as given in eq 50 is fairly general one, and this may be used for a variety of other transport studies, beyond the present application in the diffusion problem. To further extend the present work, one may, for example, use the quantum wave packet based scattering tools and solve a more accurate, perhaps a nonlinear,77 model kinetic equation. Also, we have not taken the effect of quantum statistics, arising due to the indistinguishability of particles,62 into consideration. The present work can be extended to include such effects, if required. The present work may further be extended to obtain an analytical expression for the complete diffusion tensor. The drift motion, arising due to the external electromagnetic field, is another direction where the present elaboration on transport theory may be useful. We will present such studies in future publications.

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Details of the mathematical derivations of the results presented in the paper (ZIP)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b06380. 9618


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The Journal of Physical Chemistry B

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