Up-Hill Diffusion Creating Density Gradient - What is the Proper Entropy?

N. Sato and Z. YoshidaGraduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan

(Dated: October 16, 2018)

It is always some constraint that yields any nontrivial structure from statistical averages. Asepitomized by the Boltzmann distribution, the energy conservation is often the principal constraintacting on mechanical systems. Here, we investigate a different type: the topological constraintimposed on ‘space’. Such constraint emerges from the null space of the Poisson operator linkingenergy gradient to phase space velocity, and appears as an adiabatic invariant altering the preservedphase space volume at the core of statistical mechanics. The correct measure of entropy, builton the distorted invariant measure, behaves consistently with the second law of thermodynamics.The opposite behavior (decreasing entropy and negative entropy production) arises in arbitrarycoordinates. An ensamble of rotating rigid bodies is worked out. The theory is then applied toup-hill diffusion in a magnetosphere.

There are plenty of examples that seemingly violatethe principle of entropy maximization. So-called up-hilldiffusion, creating density gradients, is often observed inmulti-phase fluids and solids undergoing spinodal decom-position [1, 2], in metallic alloys [3], nanoporous materials[4], and magmas [5]. By separating the different compo-nents of the mixture, Helmholtz free energy achieves alocal minimum, characterized by non-uniform concentra-tions, that is stable against fluctuations [2]. With a com-pletely different mechanism, astronomical plasmas accu-mulate within the magnetic fields of stars and planetsthrough the process of inward diffusion [6–9] and gen-erate an heterogeneous density profile. That the drivingforce is not the energy constraint is made apparent by theexperimental observation of non-neutral plasma particlesclimbing up the potential hill [8], as well as by numericalcalculations concerning their thermal equilibrium [10].Here, the underlying principle is the self-organization ofa quasi-stationary state, governed by long-range interac-tions, that feeds upon the topological constraints (typi-cally in the form of adiabatic or Casimir invariants [11])affecting canonical phase space. As long as the invari-ants are preserved, the ordered architecture, arising fromthe integral manifolds foliating phase space, seems to beconflicting with the second law of thermodynamics. Oncethe invariants are broken, the quasi-stationary state is de-stroyed and the systems progressively approach thermaldeath. Accretion of galaxies under the action of gravita-tion [12, 13], ferromagnetism mediated by the magneticfield [14, 15], spontaneous creation of planetary magne-tospheres through the electromagnetic interaction [6–8],vortical structures in magnetofluids preserving helicities[16], living organisms harvesting ‘negentropy’ [17], self-organization of data flows in information theory [18] aresome of the most paradigmatic examples of such orderedstructures that grow on the topological invariants affect-ing the relevant ‘phase space’.

In the present paper, we study the non-equilibriumstatistical mechanics of Hamiltonian systems subjectedto the aforementioned topological constraints. In partic-

ular, we show that the entropy defined on the invariantmeasure (the preserved phase space volume) of the sys-tem behaves consistently with the second law of thermo-dynamics. Due to the non-covariant nature of differentialentropy [19, 20], the time evolution of the uncertaintymeasured in arbitrary coordinates may ‘flip’, and appearas an entropy decrease in the Cartesian perspective. Itis the Jacobian of the coordinate change that yields theordered structure, while the probability distribution isflattened in the proper variables.

The theory, which finds its roots in the phenomenolog-ical observation that particle density in planetary mag-netopsheres tends to be homogenized in the magneticcoordinates [21, 22], shows that the proper phase spaceupon which statistical mechanics can be formulated dif-fers from the a priori variables used to represent a gen-eral physical system. These findings may pave the wayfor a new and rigorous understanding of the statisticalmechanics governing constrained systems.

We start with a short review of the Hamiltonian for-malism. Hamiltonian mechanics is the result of interac-tion between matter (energy or Hamiltonian function H)and space (Poisson operator J ) according to the equa-tion:

v = J∇H, (1)

where v = x is the flow velocity in n-dimensional phasespace. (1) admits two typologies of constants of motion:those that can be ascribed to the specific form of theHamiltonian function, i.e. to the properties of matter,and the so-called Casimir invariants that originate fromthe eigenvectors with 0-eigenvalue (the null space or ker-nel) of the Poisson operator, i.e. from the propertiesof space. This second kind of invariants, which limitsthe accessible regions of phase space as a result of theconstraining environment, is at the core of the theorydeveloped in the present work. Specifically, due to an-tisymmetry J T = −J , whenever the operator J has akernel ξ such that J ξ = 0, the system is subjected to

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topological constraints:

ξ · v = 0. (2)

(2) can be thought as the formal definition of topolog-ical constraint. We remark that the above result holdsfor any Hamiltonian, and even if J does not satisfy theJacobi identity (see [23]). However, thanks to Darboux’stheorem [24], the Jacobi identity ensures that the kernelis integrable, i.e. that a Casmir invariant exists:

ξ = λ∇C, (3)

where, for now, we assumed that the rank of J is n− 1(see [25]), and the two functions λ and C are integra-tion factor and Casimir invariant (C = λ−1ξ · v = 0)respectively.

It is now useful to make some considerations on thenon-covariant nature of differential entropy. Extensionof Shannon’s discrete entropy to continuous probabilitydistributions is a delicate process [19, 20]. Indeed, thequantity:

S = −∫V

p (x) log p (x)dV (4)

is not, in general, the entropy of the continuous prob-ability distribution p (x) on the volume element dV =dx1 ∧ ... ∧ dxn. The reason is that S is not covariant,i.e. its value changes depending on the chosen coordinatesystem, and (4) tacitly assumes that dV is an invariantmeasure. Unfortunately, this is not always the case and(4) has to be amended with Jaynes’ functional:

SJ = −∫V

p (x) log

(p (x)

I (x)

)dV , (5)

where the Jacobian I (x) compensates the coordinate de-pendence of the logarithm. In the Hamiltonian picture,one can always find a time-independent function I (x)nullifying the Lie derivative of IdV with respect to thedynamical flow v, i.e. such that LvI (x) dV = 0. Theobtained I with (5) will then give the desired covariantform of entropy. It is useful to recast (5) as below:

Σ = −∫VI

P (y) logP (y)dVI . (6)

Here, P is the probability distribution of y and dVI is theinvariant measure of the system dVI = dy1 ∧ ... ∧ dyn =IdV satisfying LudVI = 0, with u = y.

We are now ready to test the theory with a simple3D example. In 3D, equation (1) can always be cast inthe form v = w × ∇H, where w is a properly chosen

vector (see [26]). The Euler’s rotation equation for themotion of a rigid body with angular momentum x andmoments of inertia Ix, Iy, and Iz can be obtained by set-ting H =

(x2/Ix + y2/Iy + z2/Iz

)/2 and w = x. The

kernel ξ associated to this operator, i.e. the topologicalconstraint (2) affecting the phase space of a rigid body, issoon identified to be ξ = x. Indeed, ξ·v = x·x×∇H = 0.At the same time, one can verify that the Jacobi iden-tity (see [27]) is satisfied x · ∇ × x = 0, making thesystem Hamiltonian. The Jacobi identity also guaran-tees that the kernel is integrable (remember (3)) to givethe integration factor λ = 1 and the Casimir invariantC = x2/2, so that w = ∇C. Furthermore, the invariantmeasure turns out to be dVI = dx ∧ dy ∧ dz, as followsfrom ∇ · v = 0. Since this is the original statistical mea-sure, one can directly apply (4) to define the entropy ofan ensemble of such rigid bodies. However, suppose thatwe consider a slightly more complicated rotation pattern,such as:

v = λ(x)∇x2

2×∇H, (7)

where, for example, λ = ez2/2. Since z ∝ λ, high val-

ues of z will be less probable and (7) may represent theanisotropic rotation of a rigid body that tends to spinaround the axis with angular momenta x, y. (7) still sat-isfies the Jacobi identity, and thus represents an Hamilto-nian system with the same Casimir element C. However,the invariant measure becomes:

dVI = e−z2/2dx ∧ dy ∧ dz = dC ∧ dχ ∧ dz, (8)

where we introduced new coordinates (C,χ, z), with χ =

e−z2/2 arctan (y/x). Separating the constant of motion

C, the new 2D canonical equations are:

u =

[χz

]=

[−Hz

Hχ

]. (9)

One can verify that (9) is divergence free.In order to study the statistical mechanics of the new

system, we now consider an ensemble of objects obeying(9) and let them interact by adding to the Hamiltonianan interaction potential φ. Its ensemble average must gozero 〈φ〉 = 0, since the total energy of the system has tobe preserved. In addition, and this is the key point ofthe paper, there are grounds for the ergodic hypothesisin the novel coordinates (C,χ, z) (and not in the originalvariables (x, y, z)) because of the invariant measure (8).In other words, it is licit to exchange ensemble averageswith time averages only on (8):

0 = 〈dφ〉 = 〈φχ〉 dχ+ 〈φz〉 dz =

φχdχ+ φzdz = Γχ (t) dχ+ Γz (t) dz,(10)

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with Γχ and Γz Gaussian white noises and where the barindicates long-time averaging. In (10) first we substitutedensemble averages with time averages, and then repre-sented the various components with random processes ofzero time average. We remark that this would not havebeen possible in the original coordinates (x, y, z), as theyare not measure preserving. Neglecting the constant C,the equations accounting for the interaction become:

[χz

]=

[−Hz − ΓzHχ + Γχ

]. (11)

Note that, while the Hamiltonian is no more a constant,C is still a Casimir invariant: the rigid bodies will explorethe surface of phase space defined by C.

The next step is to build the Fokker-Planck equationassociated to (11). We refer the reader to [9, 29] for adetailed description of the procedure. The result is:

(12)∂P

∂t= Hz

∂P

∂χ−Hχ

∂P

∂z+

1

2Dχ

∂2P

∂χ2+

1

2Dz

∂2P

∂z2.

Here P is the probability distribution on (χ, z) and Dχ,Dz are the diffusion coefficients associated with the whitenoises. Finally, we seek for an explicit expression of theentropy production rate σ of the system. Define theFokker-Planck velocity Z to be the vector field such that(12) is written as ∂tP = −∇ · (ZP ). Then, recalling (6):

dΣ

dt=

∫VI

{P∇ ·Z +∇ · [P log (P )Z]}dVI . (13)

The first term represents the ensemble average of theFokker-Planck velocity divergence, while the second fac-tor can be cast as a surface integral representing entropyflow out L. It is straightforward to deduce that:

σ = 〈∇ ·Z〉 , (14a)

L = −∫VI

∇ · [P log (P )Z] dVI . (14b)

Substituting the expression of Z in (14a), we obtain:

σ = −1

2Dχ

⟨∂2 logP

∂χ2

⟩− 1

2Dz

⟨∂2 logP

∂z2

⟩. (15)

In figure 1 we report the results of the numerical sim-ulation of (12). The Entropy Σ, defined on the invari-ant measure (8) of the system, behaves consistently withthe second law of thermodynamics and the associatedentropy production σ is positive. On the contrary, thewrong measure of entropy S = −

∫f log fdV = Σ + 〈λ〉,

defined by the distribution function f on the originalphase space dV = dx ∧ dy ∧ dz, decreases. Furthermore,

FIG. 1. (a): Σ and S as a function of time t. (b): σ as a functionof time t. Arbitrary units are used. Initial condition is a flatdistribution f on dV .

diffusion flattens the distribution P and since preserva-tion of particle number requires PdVI = fdV , f = P/λcreates an ordered structure by approaching f ∝ λ−1.

Let us show how the theory can be applied to the studyof a real self-organizing system: a magnetosphere. Inastronomical plasmas, charged particles are trapped byplanetary magnetospheres as they spiral around the mag-netic field B = ∇ψ ×∇θ, where ψ = ψ(r, z) is the flux-function and θ the toroidal angle of a cylindrical coordi-nate system (r, z, θ). This dynamics (cyclotron motion)is characterized by preservation of the magnetic momentµ = mv2⊥/2B = const, where m is the particle mass,v⊥ the particle velocity perpendicular to magnetic fieldlines, and B = |B|. Because of the topological constraintµ, it turns out [9, 30] that the invariant measure of mag-netized particles is dVI = dµ ∧ dv‖ ∧ dl ∧ dψ ∧ dθ =Bdµ ∧ dv‖ ∧ dx ∧ dy ∧ dz = BdV , where l and v‖ arelength and velocity along B respectively. The electro-magnetic interaction diffuse the constrained particles onthe statistical measure dVI and maximize the associatedentropy Σ. Due to the inhomogeneous Jacobian B, theprocess will appear as creating density gradients and tem-perature anisotropy in the Cartesian perspective, whilethe entropy S defined on dV is minimized. This scenariois exemplified in figures 2 and 3 obtained by simulationof the Fokker-Planck equation derived in [9, 30].

FIG. 2. (a): Σ and S as a function of time t. (b): σ as a functionof time t. Arbitrary units are used. Initial condition is a Maxwell-Boltzmann distribution.

This research was supported by JSPS KAKENHI

4

FIG. 3. Self-organized plasma after entropy maximization.(a): spatial profile of particle density (a.u.). (b): temperatureanisotropy T⊥/T‖. (c): parallel temperature T‖(eV ). (d): per-pendicular temperature T⊥(eV ). White, green, and purple linesrepresent contours of B, ψ, and l.

Grant Nos. 23224014 and 15K13532.

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