Corrections to Fluid Dynamics

arXiv:math-ph/0105013v1 11 May 2001

Corrections to Fluid Dynamics R. F. Streater Dept. of Mathematics King’s College London Strand London, WC2R 2LS [email protected] 15/04/2001

Abstract We propose a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a discrete-time Markov chain for random fields on a lattice, followed by projection onto the information manifold. In the continuum limit we obtain a non-linear coupled parabolic system of field equations, showing corrections to the Navier-Stokes equations. In particular, the Euler equation for the conservation of mass acquires a diffusion term, seen to be an Ito correction; this is also the origin of the usual viscosity terms. All parameters are predicted in terms of the mass and size of the molecules. It is argued that the new equations are more stable as well as more consistent than the Navier-Stokes system.

1

Introduction

Our title is taken from Truesdell’s book [37], where he says “results of this kind are described by kinetic theorists as ‘corrections to hydrodynamics’ ”. In an earlier work [35] we tried to derive macroscopic physics from statistical dynamics [34], a kinetic theory that conforms to the laws of thermodynamics. The model in [35] involved the conservation of mass and energy, but omitted the momentum field. We set up a Markov chain on the state-space of a lattice gas of lattice size ℓ, and discrete time step dt. The diffusion limit, ℓ2 = λdt → 0 was shown to exist in the sense of means. We obtained the 1

Smoluchowski diffusion equation for the density, supplemented by a new equation for the temperature. A more general scheme has been studied by Wojnar [38]. Our system predicted the Soret and Dufour effects in liquids at rest, which might be regarded as a satisfactory result. However, without a velocity field, the theory “has not got off the ground” [25]. It is therefore worthwhile to use the same methods to construct a model with all five conserved fields, mass, energy, and the three components of momentum. These five are the constituents of the Navier-Stokes equations [27, 28]. We set up the simplest model, and find that the dynamics of the macroscopic fields can be worked out in terms of elementary functions. These can be simplified in a suitable limit, giving the modified Navier-Stokes system: ∂ρ ∂t ∂(ρe) ∂t





+ div(u(ρe + P )) = λ∇2 Θ1/2 P +

∂(ρu) ∂t



+ div(uρ) = λ∇2 Θ1/2 ρ



  3λ  4λ  1/2 ∂i ρΘ u · u + ∂j ρΘ1/2 ui uj . 5 5

+ div(ρu ⊗ u) = −∇P + +

o 2λ n 2 3∇ (ρΘ1/2 u) + ∇div(ρΘ1/2 u) 5

(1)

Here, ρ is the mass-density, e is the energy per unit mass, (not just the internal energy, as in [27, 28]), P is the pressure, Θ is the temperature, and u is the mean velocity field. The diffusion constant λ is predicted to be λ :=

ℓc , (2πΘ0 )1/2

(2)

where ℓ is the diameter of the hard core potential of a molecule of mass m, and c := (kB Θ0 /m)1/2 is the approximate velocity of sound at the reference temperature Θ0 . The limit we take corresponds to ℓ → 0, m → 0 such that ℓc remains finite and non-zero. The main difference from the usual equations is the diffusion term on the right-hand side of the continuity equation. It is this that causes the Soret effect, namely, if the density is constant but not the temperature, there will be a flow of particles. The exact form of the viscosity tensor might be model-dependent, but other terms not usually present, which express Onsager symmetry, seem to be needed for consistency. Thus the anomalous Dufour effect is the Onsager dual to the Soret effect, and appears in the 2

energy equation. These equations have the following advantages over the conventional Navier-Stokes equations. There is a natural discretisation of the system (the original model) which is a thermodynamic system in its own right. This could be the basis of a convergent numerical scheme, and is implementable if the velocities are small compared with the speed of sound. The equations are likely to be more stable, and have smoother solutions, than the equations without the diffusion terms in the density equation. So the proof of existence of solutions should be easier, as well as their numerical study. Finally, in the programme to prove the NavierStokes equations from a microscopic Hamiltonian model, either classical or quantum, by some scaling method, it is difficult to see how diffusion terms of the same size will survive in some of the equations, but not in the density equation. The Soret and Dufour effects were found experimentally in the nineteenth century. Later, they were predicted for gas mixtures by Chapman and Dootson [15]. Curiously, neither Chapman nor Batchelor [3] emphasises these effects for gases of a single type. Chapman notes that for gas mixtures ‘diffusion is produced by (1) a concentration gradient...(2) by external forces...and by variations of (3) total pressure and (4) temperature.’ For a single species, Batchelor needs to tag a particle to make his discussion, and for a mixture of marked and unmarked particles, says [3], p. 33: ‘The total number density does not itself change as a consequence of the exchange of marked and unmarked molecules and may be regarded as constant’. Thus he considers the diffusion to occur because of the exchange of particles of the same mass but differing energy etc.; no diffusion of mass occurs in the exchange. Woods agrees with this (eq. (51.6), (38.3), (53.2) and elsewhere, of [39]), but also disagrees: eq. (69.7) is Fick’s law, our first equation with u = 0 and Θ constant. Chemists [20] know all about diffusion, which is discussed on p. 11 and derived from kinetic theory in Chapter 7. This is done for gas mixtures in general, and the case of one species is obtained as a special case. This goes under the name ‘self-diffusion’. The authors say ‘Clearly, if the molecules in a gas are all physically identical, it is impossible to measure their interdiffusion’. They do claim that we can measure the diffusion coefficient by looking at the interdiffusion of isotopes or isomers (p. 540). They have in mind that the particles of one type exchange their positions with those of the other type. There is no net diffusion of mass as a whole, according to (7.2-43) of [20], in spite of its being in contradiction to (1.2-9) of the same book. It seems that for people working in fluids, the equation of continuity is sacrosanct; indeed, it follows easily from the 3

Boltzmann equation [20], or [4], page 154. Nevertheless, it contradicts the ‘well-known’ fact that a planet with a stationary atmosphere loses it eventually by diffusion into space. The quoted authors rely on the authority of Enskog [18], and Chapman and Dootson [16] who were the first to do work of this type. Ref. [15] has been criticised by Truesdell as non-rigorous, in that no distinction is made between approximations and theorems. Chapman himself has said that reading the book is ‘like chewing glass’ [13]. Truesdell’s own version [29] is limited to rigorous results, and gives what was then known about the hard problem, to show that Maxwell’s model thermalises in a small time on the macroscopic scale. He says that this result is not yet proved. The reason may be that in this work, the model attempts to describe mechanical properties of individual molecules, and probabilistic results are out of reach. A similar attitude is evident in [4]. Balescu defines the state as the specification of a phase-space distribution function F (p, q), by which he introduces probabilistic, non-mechanical concepts. He then spoils it by adding, p. 25, “the subsequent evolution of the system is strictly determined by the laws of classical mechanics” (original italics). He uses the methods of probability, and goes through the “gain-loss” equations of a Markov chain, and derives the Fokker-Planck equation. He notes that the diffusion terms are related to the singular nature of the stochastic process. But his heart is not in it; he concludes (p. 320) “As long as we admit the existence of atoms...it is impossible to deny the fact that, at each time, each particle is ‘somewhere’, at a well-defined position.” This classical model of molecules does not allow a particle to make a random move to an empty site. The crux of the question is whether the state of the system is a point ω in phase-space Ω or a probability p on Ω. The latter is chosen in statistical dynamics (the true faith, [34], not an expensive imitation). In this case for discrete systems we have the possibility of using von Neumann’s entropy S(p) = −kB

X

p(ω) log p(ω)

(3)

ω

to represent the thermodynamic entropy. Some authors deny that S(p) has anything to do with thermodynamic entropy, citing at least two reasons: S(p) is zero for any actual sample ω ∈ Ω, and S(p) is constant under any Hamiltonian dynamics. Neither criticism applies to statistical dynamics, since the state of the system is a probability and not a sample point, and the dynamics is not Hamiltonian. Another off-putting feature of the von Neumann entropy is that it diverges to ∞ as fast as − log ℓ when we take the continuum limit ℓ → 0. It can be shown [35] that in the model studied 4

there, this divergent part is time-independent, and that the time-dependent R part has a contribution of the form − ρ(x) log ρ(x) dx, where ρ(x) is the particle density in the limit ℓ → 0. This term, the ‘differential entropy’, is not always positive, and this has puzzled some authors, who are tempted to R replace it by the relative differential entropy ρ(x) log(ρ(x)/σ(x)) dx, where σ is the equilibrium density. This idea is a good one for isothermal dynamics, and is the choice of [31, 26]; however, it is related to the free energy, and is not the entropy; it does not obey a law of increase or decrease in time if the temperature is not constant. Hill says [19] that since the classical entropy is infinite, we are allowed to choose our own renormalised value so as to agree with the (finite) quantum value. Here we adopt a procedure that has this outcome, but describe it differently. We use classical probability but not classical mechanics; the latter cannot predict the size of molecules, nor the number of states per unit volume of phase-space. We introduce a discretisation of momentum, which is taken to be an integer multiple of a unit, ǫ, whose value is found from quantum mechanics. Then with ǫ and the lattice size ℓ both positive, the entropy is finite and positive, and agrees with quantum mechanics. Since these parameters are very small, we can simplify the dynamics of the model by taking the continuum limit, ℓ → 0, ǫ → 0; this allows us to replace sums by integrals and finite differences with differentials. We arrive at the system (1). In the study of equilibrium, however, we keep ǫ > 0, ℓ > 0, and can nevertheless express the equation of state in terms of elementary functions. Our programme started as a project, to derive the Navier-Stokes equations from statistical dynamics. This is less ambitious than hoping to derive hydrodynamics from the Boltzmann equation; in the latter, the non-linear collision term is thought to lead to local thermodynamic equilibrium on a short time-scale. In statistical dynamics this step is put in by hand, thus bypassing the most intractable problem in the subject. In this we follow the method of information dynamics, pioneered by Jaynes [23] and Ingarden [21, 22]. In early versions of this theory [24] the true dynamics is taken to be Hamiltonian, so the von Neumann entropy is constant in time as a state p0 evolves from time 0 to a state pt at time t. At time t the state pt is projected by a non-linear map Q onto the information manifold M of states, giving a reduced description by the state Qpt ∈ M. The entropy is thereby increased; this is interpreted as loss of information due to the observer’s giving up some detail in the projection Q. It is thus observer-dependent. It was recognised that there was a difficulty in interpreting the state Qpt as the physical state at time t; the state we get depends on how many times we 5

choose to set up the reduced description. For example, Q[(Qpt )s ] 6= Qpt+s . We do get a semigroup if we project onto the information manifold after a small time interval dt, evolve another time dt, project again, and repeat. Taking dt to zero gives the dynamics as a curve in the information manifold. This is an attractive idea, since, having made the choice of the ‘level of description’, the new dynamics is determined by the Hamiltonian. A surprising result is that this dynamics is isentropic [24, 5]. The reason is that for small distances the entropy gained by the map Q is proportional to the square of the distance (in the Fisher metric) of the point pt from the manifold, and this distance is proportional to dt. The second-order difference in entropy contributes zero to the rate of entropy gain in the limit dt → 0. The modern form of information dynamics is statistical dynamics [34]. We renounce Hamiltonian dynamics, replacing it with a stochastic process that conserves energy, mass and momentum, and also increases entropy. It can be arranged that in a small time dt, the gain in entropy is of order dt. This gain is at least preserved by the map Q, since this in an entropy nondecreasing map. We interpret this gain in entropy as real, and not observer dependent. It is the gain in entropy that would occur if the dynamics included non-linear collision terms, which involved very fast variables able to thermalise the system locally on a fast time-scale. This is equivalent to the assumption that the linear space spanned by the slow variables is, in terms used in the Boltzmann theory, a complete set of collision invariants. We regard the map Q, which is nonlinear, as part of the dynamics; it takes the place of some fast dynamics omitted from the description and implements the thermalisation of the fast variables (those random variables not in the linear span of the slow observables). It does not alter any of the means of the slow variables. It is the ‘best’ way to close the system of equations. More might be possible; in certain models, Yau [40] proves that we can permute the operations of making the reduced description with the time-evolution. Thus, we take a state and replace it by the closest macrostate, and the solve the macroscopic equations up to time t, getting another macrostate. In Yau’s models, this gives the same answer as following the exact microscopic dynamics for the same time t, and then making the reduced description. Such a result is out of reach in general classical Hamiltonian dynamics, mainly because (due to the KAM theorem) it is unlikely that such a system is ergodic. There is no such far-reaching analysis in the present paper, which is just a first step. We want to make sure that at a formal level the macroscopic equations themselves are correct. We find that the hydrodynamic equations of the model are similar to the Navier-Stokes equations, but that there are 6

some new terms of the same size as the usual viscosity terms. We argue that these terms are needed for the stability and consistency of the system. Beck and Roepstorff [6] proposed a similar programme; they start with the Ornstein-Uhlenbeck velocity process for a tagged particle, instead of Hamiltonian dynamics. They study the equations of motion for the temperature and the mean velocity field, by making use of the Ito calculus. They use the identification of temperature with mean relative kinetic energy to close the system of equations. They are able to reproduce some of the terms in the Navier-Stokes equation, but get a different viscosity term. They also find a diffusion term ‘correcting’ the Euler equation for the conservation of mass. Because this correction is rarely written down in texts on hydrodynamics, they content themselves with the remark that they agree with the Navier-Stokes equations in the incompressible case, provided that the viscosity is negligible. They stop short of saying that the continuity equation is wrong. Dobrushin [17] also obtains the diffusion term, in a simpler model than ours; he notes that it is not in the Euler equation. He says that if the Euler system has a solution, it might explain reality for a short time; but the modified equation should describe reality well for a much longer time. We concur with this sentiment. The introduction of random fields here is not the same as adding Langevin terms directly to the hydrodynamic equations. The motive for this is that it is likely that the equation (without our corrections) only has chaotic solutions; these might be describable by a random field. In §2 we specify the model, illustrating all the concepts of statistical dynamics in a very concrete way. The configuration is a field on the discrete lattice Λ, similar to that introduced in [33]; the difference is that in the present model, the local state if occupied by a particle, is specified by the three components of momentum, rather than simply by the energy. We show that the fluid at equilibrium is a special van der Waals gas, and that in constant gravity, the density falls off exponentially with height, as expected. In §3, we specify the hopping rules that give the dynamics we study in the rest of the paper. In §4, we find the hydrodynamic limit of the discrete stochastic process defined in §3, for the special case of no external field. It is written in terms of the dynamics of the means of the slow fields. We find that some terms in the the equations are not in tensor form unless we average them over the group SO(3); in other words, some of the lattice structure is visible in the preferential viscosity along the axes of the cubic lattice of the discrete model. On averaging, some of these terms are converted into the viscosity 7

tensor, and others disappear. We obtain the system eq. (1) as a formal limit. In §5 we argue the advantages of the new system over the Navier-Stokes system.

2 2.1

The model in equilibrium The information manifold

Any model of a fluid must have a local structure. In our case, space is chosen to be a lattice Λ ⊆ (ℓZ)3 ⊆ R3 , representing the possible positions of the molecules. The parameter ℓ ≈ 10−8 cm. = 1 ˚ A is the size of the hard core of a molecule, and at most one molecule can sit at each point. The sample space Ω is the set-product of Ωx as x runs over Λ; thus Ω=

Y

Ωx .

(4)

x∈Λ

The choice of the sample space Ωx at each site is part of the model; it tells us the microscopic states (momentum, spin...) available to a particle at x, if there is one. In our model, Ωx is a discrete version of momentum space, together with the empty set, signifying a hole: n

o

Ωx = ∅, (ǫZ)3 ,

(5)

where ǫ is a parameter having the dimension of momentum. A sample point is thus a function ω : Λ → such that ωx ∈ Ωx ; ω is a section of the trivial bundle Ω over Λ. Note that Ω is not a discrete version of phase space of a specific number of named particles, as studied e. g. in [29]. There, the authors adopt a ‘particle’ point of view, whereas our Ω uses the ‘field’ point of view. The number of states in a field theory is reduced by a factor (1023 )! compared with the particle theory having the same size of discretisation. Moreover, in Ω no configuration exists with more than one particle at a site x ∈ Λ; this result can only be achieved in the particle point of view by introducing an infinite repulsive core between particles, or by solving the equally hard problem of N non-intersecting random walks. The parameters ℓ and ǫ are positive and roughly predictable from quantum mechanics. There are tables of values of the size of various molecules; these values of ℓ depend on the quantum model in that, for the larger atoms and molecules the Hamiltonian chosen is itself open to change and improvement. It is better to regard ℓ as a parameter to be chosen so that the 8

predictions of the present model agree as well as possible with experiment. The corrections to Boyle’s law given in the present section would seem to be a good place to start. We shall see that we can express the partition function Ξ of the model by elementary functions, provided that ǫ is small enough so that we can replace discrete sums by integrals. To be a realistic attempt at a model of fluids, we must conform to Heisenberg’s uncertainty relations. The number of momentum states of the atom, in a region of size ℓ3 , must be equal to the number of quantum states. Otherwise the entropy, free energy etc. would differ from that of a quantum treatment, and we would not agree with experiment over the value of thermal capacities. Thus, we adopt a semi-classical treatment; we are well within the region where this is valid. The neighbouring particles provide an infinite potential well (in R1 ) of size about ℓ = 3 × 10−8 cm. The momentum k of a state is quantised and has eigenvalues ¯hκn , where κn is the wave-number κn = nπ/ℓ = 108 n cm.

(6)

So in c.g.s. units the smallest change in momentumRis h ¯ κ1 = 6.6×10−19 = ǫ, small enough to replace sums over k by integrals dk. For lower densities the size of the potential well is increased, leading to a smaller value of ǫ. This changes the absolute value of the entropy, etc., but it cancels out in the time-dependent part, if we replace sums by integrals. We forego doing the asymptotic analysis justifying this. In any statistical theory, the observables are random variables; that is, they are real functions X : Ω → R. On a countable space, every function is measurable. In our model, the ‘slow variables’ of information dynamics are the densities of the particle number, the energy and the three components of momentum along the three directions of the cubic lattice Λ: Nx (ω) =

(

0 1

Ex (ω) =

(

0 (2m)−1 k.k + Φ(x)

P x (ω) =

(

0 k

if ωx = ∅ if ωx = k ∈ (ǫZ)3

if ωx = ∅ if ωx = k.

(7) if ωx = ∅ if ωx = k

(8) (9)

Here, Φ is a real function representing an external potential. We do not solve the theory for any interactions between the particles. In the equilibrium 9

state, only the hard-core repulsion shows up. In an interesting series of papers on similar models, [7, 8, 9, 10, 11] Biler and coworkers have been able to add interaction, in the sense of mean-field theory. Each random variable divides Ω into level sets, called shells by physicists. In statistical dynamics, the shells defined by the conserved quantities play a large role. For example, the energy shells are labelled by the possible values that the total energy can have: ΩE =

(

X

ω∈Ω:

)

Ex (ω) = E .

x∈Λ

(10)

If |Λ| < ∞, and Φ is bounded below, all these energy-shells are finite sets. In the same way, the number and momentum shells ΩN , Ω̟ are defined, P P N and ̟ being the values of x Nx and x P x respectively. Thus Ω is the disjoint union of disjoint shells: Ω=

G

ΩE,N,̟

(11)

E,N,̟

where ΩE,N,̟ = ΩE ∩ ΩN ∩ Ω̟ .

(12)

The state-space of the model is the set Σ of probability measures p on Ω. If the state is p, then the macrovariables are the means of the slow variables Ex = Ep [Ex ] :=

X

Ex (ω)p(ω)

(13)

ω

Nx = Ep [Nx ] :=

X

P x (ω)p(ω)

(14)

P x (ω)p(ω).

(15)

ω

̟ x = Ep [P x ] =

X x

From these we form the densities ρ(x) = Nx /(ℓ3 ) etc. which in the limit ℓ → 0 are expected to obey hydrodynamic equations. The information manifold of our model is the subset M ⊆ Σ of states of the form p(ω) =

Y

Ξ−1 exp {−βx Ex (ω) − ξx Nx (ω) − ζ x · P x (ω)}

(16)

x∈Λ

where Ξ is the great grand partition function Ξ :=

X

exp {−βx Ex − ξx Nx − ζ x · P x } .

ω

10

(17)

If the mean fields {Ex }, {Nx }, {̟ x } are given, there is a unique state in M with these values of the means. Thus these fields, collectively denoted ηx , are coordinates for M, called mixture coordinates. The fields {βx , ξx , ζ x } obviously also determine the point p; they are called the canonical coordinates for M. By Gibbs’s principle, the state of the form eq.(16) is the state of maximum entropy among all states in Σ with the given values of the means. It follows that the map Q mentioned before cannot reduce the entropy: if p is any state (for which the means of the slow variables are all finite), we define Qp to be the unique point on M with these same means. Hence S(Qp) ≥ S(p) and EQp [X] = Ep [X ] for all slow observables X . The states p ∈ M are said to be in local thermodynamic equilibrium. They are also called macroscopic states, which is a reduced description, but which remains a state in Σ, but one which is of a special form. In the Navier-Stokes equations, the velocity field plays a major role. It is related to the above mean fields by ux = (mNx )−1 ̟x .

(18)

We shall later be able to write ux as the mean of a random field Υ x on the particle subspace Ωx − ∅. A more natural definition of u is in terms of the canonical coordinates. The means are obtained from Ξ by the usual formulae: ∂ log Ξ ∂βx ∂ = − log Ξ ∂ξx ∂ = − i log Ξ, ∂ζx

Ex = −

(19)

Nx

(20)

̟xi We can compute Ξ = Ξx = =

X

x Ξx ,

Q

i = 1, 2, 3.

(21)

where

exp{−ξx Nx (ωx ) − ωx ∈Ωx 1 + e−ξx −βx Φ(x) Z1 Z2 Z3 ,

βx Ex (ωx ) − ζ x · P x (ωx )}

(22) (23)

where Zi =

X

exp{−βx k2 /(2m) − ζxi k}

k∈ǫZ

≈ ǫ−1

Z

∞

−∞

e−βx k 11

2 /(2m)−ζ i k x

dk, .

(24) (25)

Here, ǫ = ℓ−1 π¯ h. The integrals can be done, to give Zi = ǫ−1



2mπ β

1/2

2

emζi /(2β) .

(26)

exp{−ξx − βx Φ(x) + mζ x .ζ x /(2βx )}.

(27)

Thus Ξx = 1 + ǫ

−3



2πm βx

3/2

From eq. (20) we find 2πm βx −1 = Ξx (Ξx − 1).

Nx = Ξ−1 ǫ−3



3/2

e−ξx −βx Φ(x)+mζ x ·ζ x /(2β)

(28) (29)

From eq. (21), the mean momentum ̟ is related to its conjugate canonical variable ζ by ̟i = −

∂(log Ξ) ∂ζi

2πm = −Ξ ǫ β i = −mζx Nx /βx . −1 −3



3/2

It follows that ζi = −

mζi exp {−ξ − βΦ + mζ · ζ/(2β)} β

β̟xi = −βui . mNx

(30)

We can therefore write the macrostate in terms of the more common variables, the chemical potential µ = −ξ/β

(31)

the velocity field u = −ζ/β and the temperature Θ = (kB β)−1 : p = Ξ−1 exp {− (E − µN − u · ̟) /(kB Θ)} .

(32)

Finally, we compute the mean energy in canonical variables, and then we eliminate ξ and ζ in favour of Nx and u:

12

∂ log Ξ ∂β 3/2  −1 2πm e−ξ−βx Φ(x) exp{mζ x · ζ x /(2βx )} = Ξ ǫ2 βx   3 −2 × Φ+ + (1/2)βx mζ x · ζ x 2βx   1 3 = Nx Φ(x) + kB Θ + mux · ux . 2 2

Ex = −

(33)

Thus, in these variables, the small parameter ǫ, and the large canonical variable ξ, do not show up. This is a satisfactory outcome; dividing the equation by ℓ3 , we obtain the macroscopic variable Ex /ℓ3 as the usual function of the macroscopic variables Θ, u and the mass-density ρ = mNx /ℓ3 . For everyday fluids, u · u is much smaller than 3kB Θ/m. However, there is nothing so far in the theory which is not valid for speeds as big as or greater than the speed of sound, which is approximately (mβ)−1/2 . The presence of the small term mu · u/2 in the energy will thus be important in a study of the dissipation of sound waves, or in fast flow past a stationary boundary.

2.2

Entropy and pressure

From the von Neumann formula, for a state p ∈ M the entropy is S(p) = kB

" X X

= kB

p(ω) {βx Ex + ξx Nx + ζ x · P x } + log Ξx

ω

x

X

(βx Ex + ξx Nx + ζ x · ̟ x + log Ξx ) .

#

(34)

x

Suppose that we are in equilibrium; then the fields are independent of x. From eqs. (30,31) we get ΘS(p) = E − µN − u · ̟ + kB Θ log Ξ.

(35)

Compare this with the thermostatic formula ΘS = E − µN − u · ̟ + P V

(36)

(note that the term u · ̟ is omitted in [27], eq. (1.17)) where P is the pressure and V the volume; we see that P = kB Θ

|Λ| log Ξ = kB Θℓ−3 log Ξ. V 13

(37)

P

If there are N = x Nx particles, and V0 is the smallest volume they can occupy, (one per site) then V0 = ℓ3 N and Nx = V0 /V . Also from eq. (29), Ξx = (1 − Nx )−1 = 1 + V0 /(V − V0 ). Thus at equilibrium we have the equation of state   N kB Θ V0 P = log 1 + . (38) V0 V − V0 For small V0 this is close to the van der Waals gas (P + a/V 2 )(V − V0 ) = N kB Θ,

(39)

with a = 0. Unlike the case a > 0, this model shows no failure of convexity in its isothermals. We shall see that this thermodynamic pressure is not exactly what appear in the hydrodynamic equations. This has been remarked in the literature [14, 3]; instead, the pressure appearing in the Navier-Stokes equations is called the mechanical pressure, being one third of the trace of the viscosity tensor. We confirm this in our model, though the argument given [14] is not convincing; there it is claimed that in a time-dependent situation the local fluid has no time to adjust to equilibrium, so the pressure felt at a point is not the equilibrium pressure eq. (38); this seems to be contradicted by our choice of dynamics, which forces the states to lie on M. This is not in conflict with the result, since for small densities the pressure of the perfect gas agrees with eq. (38) up to terms of order ℓ2 , the accuracy to which we work. Our thermodynamic pressure P is indeed the one whose gradient supplies the force on the liquid. Consider constant gravity Φ(z) = mgz, where we have written x3 = z. Suppose that u = 0, and ξ, β are constant. The the density of fluid in equilibrium becomes, by eq. (28), ae−βmgz . (40) N (z) = Ξ−1 ae−βmgz = 1 + ae−βmgz What is holding up the fluid between z and z + dz against gravity? It is the pressure difference between height z and height z + dz. For unit area, Pz − Pz+dz = mgNz dz × (# of lattice sites per unit volume) so −

(41)

∂P mgNz 1 ∂ log Ξz = =− 3 . 3 ∂z ℓ βℓ ∂z

Since log Ξ satisfies the boundary condition (zero at z = +∞) we must have P = ℓ−3 β −1 log Ξ ≈ N kB Θ/V 14

for small V0 /V . For any state in M, whether in equilibrium or not, the local pressure at a site x ∈ Λ may be defined as Px = ℓ−3 kB Θ log Ξx , which in the same approximation is Px = ℓ−3 Nx kB Θ.

3

Hopping dynamics

We complete the definition of our model by giving the hopping dynamics, also called the update rules. In the general scheme which we call statistical dynamics the linear part of the dynamics, the Markov transition T , can be any bistochastic map which conserves the desired conserved global observables. It is known that if a stochastic map increases the entropy of any state (or leaves it the same) then it must be bistochastic; therefore, by Birkhoff’s theorem, [1] it is a mixture of permutations. In our case, then, we could choose T to be any mixture of permutations of ΩE,N,̟ , with no obligation to impose any relation between the operators on different shells. This gives us wide freedom, too wide, to construct models obeying just the first and second laws of thermodynamics. To get the usual equations for fluids, other physical properties need to be taken into account. That T should be local, coupling only nearest neighbours, is one natural requirement. Moreover, in a transition from one configuration (at time t) to another (at time t + dt), not only should we remain in the same shell ΩE,N , but the net rate and direction of the transfer of energy and density to and from x ∈ Λ should correspond to the velocity of the particle at x and its neighbours. When Φ = 0 this is vx (ω) := kx /m. This velocity might not take the particle at x to another site in Λ in exactly dt. We want more; it should jump (if it does jump) to a nearest neighbour. We achieve this in the present model by replacing the deterministic motion by a stochastic map. This is a combination of hopping maps to nearest neighbours along the vectors ℓ(e1 , e2 , e3 ). Given that x ∈ Λ is occupied, and x + ℓei is empty, the hopping rate is vxi /ℓ if vxi > 0, and zero if it is negative, in which case, the particle has a probability −vxi /ℓ of going to x − ℓei . The energy, mass and momentum is transferred from x to x + ℓei , and the updated state has the site x empty, and the site x + ℓei occupied with a particle of momentum k. The probability of no transition P in time dt is then 1 − ℓ−1 dt 3i=1 |vxi |. If the external field is not zero, then speaking classically the particle slows down or speeds up as it makes the jump. Suppose for definiteness,

15

Φ(x + ℓei ) > Φ(x); then if 1/2



kxi ≥ 2m(Φ(x + ℓei ) − Φ(x))

(42)

it has enough energy to reach x + ℓei ; its momentum on arrival will be kxi′ , where kxi2 − kxi′2 = 2m(Φ(x + ℓei ) − Φ(x)). (43) We take the rate of transition to be the average of the initial and the final rates: vxi := (kxi + kxi′ )/(2mℓ). (44) Then not only is the total mass and energy conserved, but the local rate of transfer of momentum obeys Newton’s law: dk dx = (k′ − k)ℓ = (k′ − k)v dt = (k′ − k)(k + k′ )dt/(2m) = −(Φ(x + ℓei ) − Φ(x))dt ∂Φ dx dt. = − ∂x Hence ddtk = −∇Φ. If kxi > 0 but it fail to satisfy eq. (42), then classically the particle returns past x without reaching x + ℓei . Let us take it that it sticks at x, so that no transition in the i-direction occurs. So we could define vx to be zero in this case. If kxi < 0, and Φ(x) > Φ(x − ℓei ) is assumed, then the particle will arrive at x − ℓei with the (negative) momentum kxi′ given by kxi′2 = kxi2 + 2m(Φ(x) − Φ(x − ℓei )). Again, the rate is taken to be the average of the initial and final rates, and this obeys Newton’s law. In each case, the transition is from a point ω that has x occupied with momentum kx and x ± ℓei unoccupied, making a jump with rate v/ℓ to the point with x unoccupied, and x ± ℓei occupied, with momentum ki′ . This jump occurs for each k, and so a transition occurs with the sum of all these rates, which increase with |k|. This will diverge to infinity, unless we limit the maximum momentum with a cut-off, |k| ≤ K say, making the sample space finite. Let Txi denote this transition matrix. We must add similar transitions for each point x ∈ Λ. Then we must choose dt so small that the sum of all hopping probabilities is less than, say, 1/2, leaving a reasonable stay-put probability (for the rapid numerical convergence of the iterated up-date). Let T denote this sum. 16

It is not reasonable to try to construct a symmetric Markov chain T for our model. For, a particle at x + ℓei with positive momentum kxi′ , classically would go on to x + 2ℓei , and should not be given the same probability of returning to x as was given to the particle with momentum kxi hopping from x to x + ℓei . The rules we have given do assign this same transition probability to a particle at x + ℓei , with the momentum −kxi′ , to jump to x, but this does not lead to a symmetric Markov matrix. We are therefore in danger of not satisfying the second law. It is likely that we can express our map as a mixture of permutations of the sample space, and so would be bistochastic. We have not pursued this, as it means deciding on rules for many-body collisions, which is a problem we wish to avoid. We are thus obliged to check the positive definiteness of entropy production in the continuum limit. For a component of Txi to be non-zero, the site x must be occupied and the site x + ℓei empty. So as usual, the entries of the Markov matrix are conditional probabilities. The rate of transition is therefore multiplied by Nx (1 − Nx+ℓei ). This is a kind of Fermionic factor, reflecting the existence of the hard core, and is worked out in [35, 36] for models without velocity. There, it is seen that if we neglect Nx compared with 1, then we still get a model in which both laws of thermodynamics hold. For simplicity, we leave out these terms, and so get a system which, like that of Navier and Stokes, is linear in the density. The hard core terms are similar to those of the Fujita model, except that they have opposite sign, and so tend to stabilise rather than destabilise the heat equations.

4

Hydrodynamics in Zero Field

If the external field Φ is zero, the momentum is conserved, and the hopping rules become much simpler, since k′ = k in that case. Thus we take the hopping rate from x in the direction ei to be kxi /(ℓm). Another simplifying fact is that if Φ = 0 then the energy-momentum leaving the site x at the start of a hop is equal to that arriving at x + ℓei at the end of the hop. Let Xx be one of the random fields N , E, P. For each i, in unit time, the change in Xx := Ep [Xx ] due to T i has four terms, which are the gain/loss to x + ℓei , and the loss/gain to x − ℓei . The change in Xx involving x − ℓei is Jxi /ℓ, where Jxi = −

X

m−1 px (k)|ki |Xx (k) +

k:k i 0

17

m−1 px−ℓei (k)ki Xx (k).

(45)

Then the change due to exchanges with both x ± ℓei is 



i i δXx = −ℓ−1 Jx+ℓe i − Jx .

(46)

That this is a finite difference means that the total amount of X is conserved (in the mean). When we take the limit ℓ → 0, and sum over i, eq. (46) becomes the divergence of the vector J. However, we are taking the limit ℓ → 0 subject to keeping the term ℓc finite, where c is the large parameter c = (β0 m)−1/2 = (kB Θ0 /m)1/2

where Θ0 is a reference temperature;

c is roughly the speed of sound at temperature Θ0 . To be sure of using this limit consistently, we keep the next term in the finite difference: 



i i ℓ−1 Jx+ℓe = i − Jx

∂2J i ∂J i + (ℓ/2) 2 + O(ℓ2 ). ∂xi ∂xi

(47)

For a given p ∈ M ⊆ Σ(Ω) suppose that Nx = Ep [Nx ] is not zero. We can then define the local probability p¯x on Ωx − ∅ by p¯x (k) = (Z1 Z2 Z3 ǫ3 )−1 exp{−βx |k|2 /(2m) − ζ x · k}.

(48)

The mean of a random variable X on Ωx − ∅, Ep¯[X] is then the conditional expectation Ep [X|Nx = 1]. Here, we define X as a random variables on Ωx − ∅ by restriction to this (smaller) sample space, which is the same for all x. So u becomes the mean of the random variable Υ := m−1 ̟/Nx , where Nx is a sure function. Then we can rewrite Jxi as Jxi = Nx Ep¯[m−1 ̟X] − (ℓ/m)



X 

ℓ−1 (Nx p¯x − Nx−ℓei p¯x−ℓei ) X(k).

k:k i >0

(49) The first term contains the product of the first moments, which gives the Euler flow Nx ux Ep¯x [X] = ux Xx , as well as the correlation between Xx and u; this will be denoted by Ep¯x [uX]T , meaning the truncated part. The second term of eq. (49) is of order ℓ, and so is negligible unless it contains a factor c. We can therefore replace the finite-difference by a derivative. Moreover, we replace the sum by an integral. Thus we get Jxi

=

uix Xx

∂ + Nx Ep¯x [u X]T − (ℓ/m) ∂xi i

18



Nx

Z

ki ≥0



ki p¯x (k)X(k)dk1 dk2 dk3 . (50)

With this form for J i we can sum over i = 1, 2, 3. We shall see that for some choices of X it is not in tensor form. This means that some remnants of the orientation of the lattice survive, violating rotation covariance. In those cases, we must average the dynamics over all orientations of the lattice. This will average will be denoted by h i. Putting this together, the equation of motion of Xx becomes *

X ∂2J i ∂Xx + div hJi = −(ℓ/2) ∂t ∂x2i i

+

.

(51)

In the analysis below, the r.h.s. of this equation will be called ‘the unwanted term’. The Navier-Stokes equations in [27, 28] are written in terms of ρ(x, t) = mNx /ℓ3 , e = Ex /mNx and u = ̟/mNx , and it is a simple matter to eliminate the temperature Θ from our equations, by using eq. (33) in order to compare our equations with those of [27, 28]. Note that in [27, 28] e differs from our e by the kinetic energy.

4.1

Corrected Euler equation for the density

We put Xx = Nx in eq. (50), noting that N = 1 on Ωx − ∅. We see that the ‘unwanted’ term on the r. h. s. of eq. (50) does not acquire a large factor c, and can so be discarded. Moreover, since N is the identity on Ωx − ∅, the correlation term does not arise. The particle current is therefore 

Jxi = ui Nx − (ℓ/m)∇i Nx

Z

ki ≥0



p¯x (k)ki dk1 dk2 dk3 .

This is evaluated in the Appendix, and gives the equation of motion for ρ(x) := mℓ−3 Nx : ∂ρ + div jN = 0, (52) ∂t where jN



:= uNx − (2πΘ0 )−1/2 (ℓc)∇ Θ1/2 Nx − +



1/2  Θ0 ℓc n 2  −1/2 2 ∇ ρΘ (u · u/c ) 10(2π)1/2



2∂i ∂j (ui uj /c2 )ρΘ−1/2

19

o

.

(53)

It is convenient to write

ℓc . (2πΘ0 )1/2 Then to order ℓc, the correct mass-current is λ :=

(54)





jc := mJN /ℓ3 = ρu − λ∇ Θ1/2 ρ .

(55)

Even up to hurricane winds the terms involving u/c are small. Near a boundary the gradients of such flows can be as big as any other term in the dynamics. However, at a boundary we must modify the hopping rules, so that in the outward normal direction the velocity of all particles is reversed in one time-step, with a suitable rate. It is not clear how to translate this idea into boundary conditions in the continuum. In the Euler system, one requires that u should be parallel to the boundary; in the Smoluchowski system, one imposes Neumann conditions on the diffusion operator. In the Navier-Stokes system, it is common to impose the condition u = 0 on the boundary, with the idea that some fluid adheres to the boundary and is at rest. In our system, one might impose that u vanish as we approach the boundary, to avoid infinite viscosity. The idea of adhesion would lead us to expect that a boundary layer is formed, and that it can widen, or evaporate, so that the normal component of the diffusion current might not be zero.

4.2

Corrected equation for the energy

The contribution of Ti to the time-derivative of X = E is ∂Ji ∂Ex =− ∂t ∂xi

(56)

because the ‘unwanted’ term on the right of eq. (51) is odd and its average over SO(3) is zero. From eq. (51), we have for J1 J1 = Nx Ep¯ "

k1 m



̟1 ∂ Ex − ℓ m ∂x1 

k12 + k22 + k32 2m

!



Nx Zǫ3

Z

∞

−∞

dk2 dk3

Z

o

∞

dk1 #)

β 2 (k + k22 + k32 ) − ζ · k exp − 2m 1 

The first term is Nx Ep¯



̟1 Ex m



̟1 ̟1 = Nx Ep¯ Ex Ep¯ [Ex ] + Nx Ep¯ m m Nx ∂ 2 log Z . = u1 Ex + m ∂ζ1 ∂β 



20





T

.(57)

Now, Z = Z1 Z2 Z3 and by eq. (26), log Zi = const. + 1/2 log β + mζi2 /(2β).

(58)

The we use eq. (30) to eliminate ζi to get Nx Ep¯



̟ Ex = uEx + Nx kB Θx u. m 

(59)

The first term is the transport and is part of the  of energy  by convection,  “correct” expression jN e = uρ − λ∇ Θ1/2 ρ e. The second term is the ideal-gas approximation to the pressure, and added to the first, it converts the transport of energy into the transport of “enthalpy” 5 m kB Θ + u · u 2 2 [14]. In our model, the limit ǫ → 0 ℓ → 0 leads to the perfect gas, and we identify ℓ−3 Nx kB Θ with the pressure. The diffusive contribution to eq. (57) has three terms, M1 =

Z

∞

0

(

βk2 k1 exp − 1 − ζ1 k1 2m −1

(Zj ǫ) M3 =

Z

0

∞

Z

∞

−∞

k13 exp 2m

kj2 exp 2m (

(

)

−βkj2 1 1 − ζj kj dkj = kB Θ + mu2j , 2m 2 2 )

j = 2, 3.

)

−βk12 − ζ1 k1 dk1 . 2m

The contribution to the diffusion current of the energy is then i ℓ ∂ h −1 N (Z ǫ) M (ζ ) x 1 3 1 2m2 ∂x1    1 2 1 2 ℓ ∂ −1 (Z1 ǫ) Nx M1 (ζ1 ) kB Θ + u2 + u3 . m ∂x1 2 2

− −

(60)

We first evaluate the contribution independent of the velocity: we ignore the velocity terms u2j and evaluate M1 (0) = m/β, M3 (0) = 2(m/β)2 and ǫZ1 (0) = (2mπ/β)1/2 . In this approximation, the diffusion current is 1/2 1/2 β ℓ ℓ β m 2 m N + −∇ 2 Nx 2 x 2m2 2πm β m 2πm β n o n o 2ℓckB 3/2 3/2 ∇ N Θ = −2λk ∇ N Θ . = − x B x (2πΘ0 )1/2

(







21







)

This contains

  3 − kB Θλ∇ Θ1/2 Nx , 2 which is the convection of internal energy due to the diffusive particle current; the remaining 1/2 is called the anomalous Dufour effect [35]. In Appendix 5 we derive the viscosity corrections to the energy current, up to quadratic powers of the velocity, which gives the rest of the energy equation in (1).

4.3

Corrected dynamics of the momentum field

Put X = P j in eq. (51) and consider the dynamics of the component ̟j due to the transition matrix Ti . The convective part of the current is then mNx Ep¯[Υi Υj ] = mNx Ep¯[Υi ]Ep¯[Υj ] + Ep¯[Υi Υj ]T ∂2 = mNx ui uj + m−1 Nx log Z ∂ζi ∂ζj = mNx ui uj + Nx kB Θδij .

(61)

Thus we get two terms of the Navier-Stokes equations, the second again being the perfect-gas approximation to the pressure. For the diffusive part, the cases i = j and i 6= j are different. If i 6= j, then the ‘unwanted’ term on the right-hand side is of order ℓ and is not multiplied by c, and so is discarded. There remains the contribution to i : JP j

ℓ ∂ − Nx (Zǫ3 )−1 ki dki m ∂xi ki >0 i ℓ ∂ h = − (Zi ǫ)−1 M1 (ζi )̟j . m ∂xi 

Z

Z

kj dkj

Z

βk · k −ζ ·k dkj ′ exp − 2m 



(62)

We can interpret the two parts:   i ∂̟j ℓ ℓ ∂ h −1 −1 (Zi ǫ) M1 (ζi ) . (Zi ǫ) M1 (ζi ) ̟j − − m ∂xi m ∂xi

The first can join the convective term Nx ui ̟j to form the convection of momentum by the complete mass current jci . The second term is part of the diffusion current −

i ∂̟ ℓ ∂ h j , (Zi ǫ)−1 M1 (ζi ) m ∂xi ∂xi

22

j 6= i.

The expression with M1 at ζi = 0 is covariant when we add a similar term from j = i; we argue in Appendix 6 that the higher powers of ζi , which must be averaged over SO(3), are negligible. To first order in u, since M1 (0) = m/β (Appendix 1), we have "

i JP = −ℓ∂i Nx j



m 2πβ

1/2

#

uj = −

  ℓ4 c 1/2 ρΘ u , ∂ i (2πΘ0 )1/2

j 6= i.

(63) There remains the term with j = i. This time, the unwanted term of eq. (51) is not negligible, and must be included. We note that h

i

Ep¯ P 2i = (Zi ǫ)−1 (M2 (ζi ) + M2 (−ζi )) .

(64)

i to be: We find the total contribution to the diffusive current JP i

"

= =

#

h i ∞ βk2 ℓ ℓ ∂i Ep¯ Nx P 2i − ∂i Nx (Zi ǫ)−1 − ζi k} dk k2 exp{− 2m m 2m 0    ℓ −1 1 ∂i Nx (Zi ǫ) (M2 (ζi ) + M2 (−ζi )) − M2 (ζi ) m 2 h i ℓ (65) ∂i Nx (Zi ǫ)−1 (M2 (−ζi ) − M2 (ζi )) . 2m Z

To order ζ, this is 2ℓ ∂i m

"

β 2πm

1/2

2

Nx m ζi /β

2

#

"

= −2ℓm∂i Nx



β 2πm

1/2

#

ui /β .

Half of this is the missing j = i term of the covariant expression eq. (63), and the rest gives a further term of order u, −ℓc



1 2πΘ0

1/2

i

h

i

h

∂i mNx Θ1/2 ui = −λ∂i mNx Θ1/2 ui .

This is not covariant and averaged over SO(3) it gives zero. But it is supposed to be a contribution to a vector, and naturally, its average is zero. To i by taking its find out what vector it is, we first make a scalar out of JP j divergence and contracting it with an arbitrary constant vector bj ; then we write the average of this scalar as bj h∂i JPi j i. Since only j = i contributes to this term, we must average the scalar 1 −ℓc 2πΘ0 

1/2

h

i

∂i2 mNx Θ1/2 ui bi , 23

and by Appendix 3, this is i λ h − ∂i ∂i (mNx Θ1/2 uj bj ) + 2∂j (mNx Θ1/2 ui bj ) . 5

We can remove the vector b and write mNx /ℓ3 = ρ to get the contribution to the current of momentum density −

i λh ∂i (ρΘ1/2 uj ) + 2∂j (ρΘ1/2 ui ) . 5

This joins the current from eq. (63), supplemented by the term i = j, to give eq. (1).

5

Conclusions

The Boltzmann equation directly leads to the Euler law of conservation of mass, without our diffusion terms [4], p. 154; it can lead to viscosity terms in the Navier-Stokes equations for the momentum [15], so it seems clear that in any derivation of the Boltzmann equation, some terms of the size ℓc are dropped, while some are kept. One reason for the ambiguity might be because the particles are treated as points from the start, but are then given a non-zero scattering cross-section. Another is the assumption made from time to time when working with the Boltzmann equation that the particles are actually at some point with a certain velocity, at each time, rather than being described by a probability. The variance of a sure function is zero, so the diffusion terms will never be derived rigorously from such an approach. Our scheme is to try to “carve nature at its joints” [2], p 341. By ignoring the particle-particle attraction we can compute the states of local thermodynamics equilibrium exactly. In a more detailed analysis, the effect of the hard-core can be catered for by the exclusion of more than one particle on each site. This shows up as a non-linear term ρ(1 − ρ/ρmax ) [36]. The external potential does not alter the local state, but does affect the hopping rates, and thus appears in the equations of motion. In a paper in preparation we find the equations of motion for a fluid moving in a potental; when u = 0 they reduce to those found for the density and energy, except for phase-space factors due to the different sample space [35]. It is possible to extend the theory to the case of inter-particle potentials by following a suggestion of Biler and collaborators [7, 8, 9, 10, 11, 12, 30]. This gives a microscopic dynamics in which the rate of hopping is governed by the mean 24

field of all the others. This is not too bad, since anyway after the map Q only the mean energy etc is conserved at the macroscopic level. A prize has been proposed for the satisfactory answer to the existence problem for the Euler equations. Since this involves the pressure, and the perfect gas equation involves the temperature, and the concept of temperature involves randomness, it is likely that the Euler equations are badly posed physically and possibly mathematically. It would be better to propose a prize for the solution to the corrected equations, as in the system (1).

6

Appendices

6.1

Appendix 1

Let Mn (ζ) =

Z

∞

kn exp{−βk2 /(2m) − ζk}dk,

n = 0, 1, 2, 3.

0

Then





M0 = (2πm/β)1/2 exp{mζ 2 /2β}Erf c (m/β)1/2 ζ , and

∂M0 ∂ζ ( )  3/2   mζ 2 m 1/2 m − (2π) ζ exp Erf c (m/β)1/2 ζ β β 2β m (1 − ζM0 (ζ)) . β

M1 = − = =

Write M0 = a0 +a1 ζ+a2 ζ 2 +. . .. Then the identity eq. (66) gives immediately a0 = 1/2(2π)1/2 (m/β)1/2 a1 = −m/β a2 = 1/4(2π)1/2 (m/β)3/2 a3 = −1/3(m/β)2 a4 = 1/16(2π)1/2 (m/β)5/2 a5 = −1/15(m/β)3 .

25

Hence m (2π)1/2 m3/2 m 2 2 ζ + − ζ . β β 2β 3/2  2  3/2  5/2 m 3 1 m m − 2 ζ + ζ2 1/2 1/2 β β β 2(2π) 4(2π) 

M1 = M2 =

M3 = 2

6.2



m β

2

−

3(2π)1/2 2



m β



5/2

ζ+4



m β

3

(66) (67)

ζ 2.

(68)

Appendix 2: the Fick term −1

(Z1 ǫ)

Z

∞

0

βk2 − ζk} = k exp{− 2m



β 2mπ

1/2

exp{−mζ 2 /(2β)}M1 (ζ). (69)

We evaluate this up to degree 2 in ζ; we get 

β 2πm

1/2 

m (2π)1/2 − β 2



2

1 − mζ /(2β)



m β

3/2

ζ+



m β

2

ζ

2

!

.

The odd terms vanish after averaging over SO(3). To order ζ 2 we get 1 m1/2 + 2 (2πβ)1/2



β 2πm

1/2

(m/β)2 ζ 2 .

(70)

The Fick term is therefore ∇

2

ℓ ρ (2πmβ)1/2





= =

6.3

ℓ kB Θ 1/2 2 ∇ ρ m (2π)1/2   ℓc 2 1/2 ∇ Θ ρ . (2πΘ0 )1/2 



!

Appendix 3: Averaging over SO(3)

The hopping transition along the directions 1, 2, 3 often gave us a contribution to a scalar quantity of the form A=

X

∂i ∂i (ui vi ) =

X

Ai , say,

(71)

i

i

where u, v are vector fields. Since this is not a scalar, the dynamics shows up the lattice structure at the accuracy where these terms enter. To remove the lattice effect, we average over the group G = SO(3) of orientations of 26

the lattice. The following is an elementary way to do this. First, let G0 ⊆ G be a subgroup and let h•iG0 denote the average over G0 . Then X

h

Ai iG0 =

X

hAi iG0 ,

i

i

and hhAi iG0 iG = hAi iG . We now find the average of A1 over the subgroup G3 of rotations in the 1 − 2-plane. After a rotation through an angle θ, ∂1 → ∂1 cos θ + ∂2 sin θ, and the same for u1 and v1 . Hence hA1 iG3

1 2π = dθ (cos θ∂1 + sin θ∂2 )2 2π 0 (cos θu1 + sin θu2 ) (cos θv1 + sin θv2 )  3 2 ∂1 u1 v1 + ∂22 u2 v2 = 8  1 1 2 + ∂1 u2 v2 + ∂22 u1 v1 + ∂1 ∂2 (u1 v2 + u2 v1 ). 8 4 Z

We now add the terms got by 1 → 2, 2 → 3 and 1 → 3, 2 → 1, which we get from ∂22 u2 v2 by averaging in the 2-3 plane, and from ∂32 u3 v3 by averaging in the 3-1 plane. Thus hA1 iG3 + hA2 iG1 + hA3 iG2 = + +

 3 2 ∂1 u1 v1 + ∂22 u2 v2 + ∂32 u3 v3 4

 1 2 ∂1 (u2 v2 + u3 v3 ) + ∂22 (u3 v3 + u1 v1 ) + ∂32 (u1 v1 + u2 v2 ) 8 1 (∂1 ∂2 (u1 v2 + u2 v1 ) + ∂2 ∂3 (u2 v3 + u3 v2 ) + ∂3 ∂1 (u3 v1 + u1 v3 )) . 4

Now, ∂i ∂j uj vj and ∂1 ∂j ui vj are invariants, and 



∂i ∂i uj uj = A + ∂12 (u2 v2 + u3 v3 ) + ∂22 (u3 v3 + u1 v1 ) + ∂32 (u1 v1 u2 v2 ) , and

∂i ∂j ui vj = A+(∂1 ∂2 (u1 v2 + u2 v1 ) + ∂2 ∂3 (u2 v3 + u3 v2 ) + ∂3 ∂1 (u3 v1 + u1 v3 )) . Hence hA1 iG3 + hA2 iG1 + hA3 iG2 =

 1 3A 1  2 + ∂i uj vj − A + (∂i ∂j ui vj − A) . 4 8 4

27

Average over SO(3). The left-hand side is hAiG , and the right-hand side is 3A 1 2 1 + ∂i uj uj + ∂i ∂j ui vj . 8 8 4 It follows that 1 (72) hAiG = (∂i ∂i uj vj + 2∂i ∂j ui vj ) . 5 As expected, this is the divergence of something, namely 1 (∂i uj vj + 2∂j ui vj ) . 5

6.4

Appendix 4: Velocity corrections to the diffusion current

From eq. (70), the terms of order ζ 2 in the diffusion current of the density is  1/2 β 1X 2 ∂ m2 u2i 2 i i 2mπ

since βu = ζ. By eq. (72), when averaged over the rotations, this makes a contribution to the diffusion density current equal to −

     m2 1 ℓ −1/2 −1/2 N (k Θ ) u u N (k Θ ) u · u + 2∂ ∂ x B x i j x B x j i 2 5mℓ3 (2πm)1/2

ℓ = − 10



m 2πkB

1/2 

Put c = (kB Θ0 /m)1/2 . Then we get the contribution −

6.5



∂i (ρΘ−1/2 u · u + 2∂i ∂j (ρΘ1/2 ui uj ) .

1/2  Θ0 ℓc  −1/2 2 −1/2 2 ∂ (ρΘ u · u/c ) + 2∂ (u u ρΘ /c ) . i j i j 10(2π)1/2

Appendix 5: Velocity corrections to the energy current

The diffusion term in the energy-current due to T1 is  Z Z ∞ Z ∞ ℓ ∂ J1 = − dk2 dk2 dk1 Nx (Zǫ)−1 m ∂x1 −∞ 0    k1  2 βk · k 2 2 −ζ·k k + k2 + k3 exp − 2m 1 2m " ) (  1/2 β ℓ ∂ mζ12 = − Nx exp − m ∂x1 2πm 2β (

M3 mu · u mu21 + M1 kB Θ + − 2m 2 2 28

!)#

.

The part of this independent of u was found in the text. The part linear in u is odd, and averages to zero. The part quadratic in u is, by Appendix 1, the quadratic part of "

β ℓ ∂ Nx = − m ∂x1 2πm +



m m2 ζ12 + β β2

ℓ ∂ = − m ∂x1

"

!

1/2

mζ12 1− 2β

!(

mu · u mu21 − kB Θ + 2 2

β 2πm

1/2

!)#

!

.

2m2 u21 m2 u21 − β 2β

m2 u · u m2 u21 mζ12 mkB Θ + m + − − 2β 2β 2β 2     3λ λ ∂1 ρΘ1/2 u21 . = − ∂1 ρΘ1/2 u · u − 2 2 2

2m2 m3 ζ12 + 4 β2 β3

1 2m

kB Θu21

!#

,

Averaging over SO(3) gives for the part of the energy-current quadratic in the velocity:  3λ       λ  1/2 ∂i ρΘ u · u − ∂i ρΘ1/2 u · u + 2∂j ρΘ1/2 ui uj . 2 10   3λ  4λ  1/2 = − ∂i ρΘ u · u − ∂j ρΘ1/2 ui uj . 5 5

−

6.6

Appendix 6: Velocity corrections to the momentum equation

The contribution to the momentum current JP j from Ti is − and from j = i is

i ℓ h ∂i Nx (Zi ǫ)−1 M1 (ζi )̟j , m

i ℓ h ∂i Nx (Zi ǫ)−1 M2 (ζi ) . m For j = i the unwanted term is

−

h h i i ℓ ℓ ∂i Ep¯ P)i2 = ∂i Nx (Zi ǫ)−1 (M2 (ζi ) + M2 (−ζi )) . 2m 2m Adding these, the contribution to the current from Ti when j = i is h i ℓ ∂i Nx (Zi ǫ)−1 (M2 (−ζi ) − M2 (ζi )) . 2m

29

To get a scalar, we take the divergence and form the scalar product with a constant vector b, to form the contribution from Ti to ∂i2 JPj bj . We get 

ℓ β − ∂i2 ( m 2πm =



"

ℓ 2 ∂ 4Nx 2m i



1/2

β 2πm

mζi2 1− 2β

1/2

1−

!

mζi2 2β

m + β !



m β

2

ζi2

m3

m2

!

X j6=i

!



̟j bj  #

2 + ζ 2 ζ i bi . 2 β 3 β3 i

Now use ̟j = Nx muj and ζi = −ui /β. Terms linear in ζi give us the expression  

β ℓ − ∂i2  m 2πm

1/2



"

X β m2 2ℓ uj bj  − ∂i2 Nx Nx β m 2πm j6=i 

1/2

m2 ui bi β

#

which was used in the text, and leads to the velocity equation in the system (1). Terms quadratic in ζ are odd, and average to zero. Terms cubic in ζi lead to a scalar that is the contraction of a sixth order tensor. We shall not evaluate this, as it is of size ℓ/c and therefore negligible. Indeed, the contribution to the scalar at third order in ζ is "

β ℓ 2 ∂ Nx 2m 2πm 

1/2 (

8m3 3 m2 ζ 4 2ζ + β 3β 3

)

m 2 ζ 1− 2β

#

.

The ζ 3 term is, up to constant factors "

ℓ 2 m ∂i Nx m β 

5/2

ζ

3

#

=

i ℓΘ0 2 h −1/2 ∂i ρΘ u⊗u⊗u c

and this is negligible in the limit ℓ → 0 and ℓc remaining fixed.

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