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KITAHARA, Kazuo

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TitleA note on Description of Hydrodyanmic Fluctuation by aMaster EquationAuthor(s)KITAHARA, KazuoCitation物性研究 (1984), 41(6): 529-533Issue Date1984-03-20URL http://hdl.handle.net/2433/91215RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA Note on Description of Hydrodynamic Fluctuationby a Master EquationA Note on Description of Hydrodynamic Fluctuation by a Master, EquationKazu 0 KITAHARAThe hydrodynamic equations describe evolution of local density, local momentum densityand local energy denstiy. For example, Fick's law for local temperature T(f, t),:t T(f, t) = K V2 T(f, t) (1)describes the diffusion of in ternal energy. This local temperature T(f, t) is a coarse-grained quantity.If we make finer measurement of the local temperature, it is fluctuating and we may look uponEq. (1) as an equation for the averaged quantity <T(f, t». In this note, we discuss howto estimatethe temperature fluctuation <oT(f, t) oT(f', t» in non-equilibrium conditions.First, we review works done in this direction by van Kampen(1972), and by Nicolis & Malek-Mansour(1983). We divide the system into many local cells. Internal energy of a local cell, whichis located at r, is denoted by Er. We consider the probabilityP({E},t)=P(E;, Ey: ..... ,t)1 2(2)of having internal energy E-; ,Ey , ... , in cells at'l ,'2, ... , at time t. Assuming Markovian process1 2for change of internal energy, we write the master equationaatP( {E} , t) =fd€ L W;(E; + € , E;+f - c -+ Er, Ep+f)r,1x P( . . . ,Er + c , Ep+l - c , . . . ,t)x P( .. , , Er, Ep+l' ... , t).-529-(3)KITAHARA KazuoThe transition probability W-r(Er+c, Er+l-c~E;, E;+I) should satisfy the following conditions,I) Detailed balance at equlibrium2) Equilibrium probability distribution is related to local entropyPeq ({E}) = exp {k~ f S; (E; )}.We choose the transition probability of the following form,== ¢( c) exp {- ~B [S; (E; + c ) - S; (E; )] } .Then, expanding the expression in c , we obtain~ ¢ ( c ) exp ( - c /kB T; ),where T; is local temperatureExpanding the master equation in c , we obtain--.L P ( {E}, t ) = ~~~ (L - ~ ~ ) f (E; ) P ( {E}, t )at r,[ aE; aE;+l .-530-(4)(5)(6)(7)(8)(9)whereA Note on Description of Hydrodynamic Fluctuationby a Master EquationHencef (E; )=fde ¢ (e) e exp( -e /kBT;)Q(E;)=fde ¢(e) e2 exp(- e/kBT;).Q (E;) = _ af(E;)a(1/kB T;:)(10)(11)(12)For small fluctuation, we may derive from the drift term of Eq. (9), the average evolution,-aa <E;:>~- ~ [f«E;:»-f«E;:+I»]t 1(13)where I is the size of cells (the mean free path). Since the size of the cell is fixed, we may writed<E;> = Cvd<Tr> from thermodynamics. By comparing Eq. (13) with the Fick's law (1), weobtain(14)This determines Q(Ep) by the relation (12), namely(15)When there is convection, the description is more complicated because we cannot fix eachcell at a given position. Cells are moving by the convection. We assign a vector ato a cell. Theposition of a cell a at time t is denoted by x(ii, t). Thenlr ; (ii, t ) =ii (X (ii, t ), t)x(a, 0) =a-531-(16)KITAHARA Kazliowhere u(x, t) is velocity field at x at time t. Let us denote internal energy and momentum of a cellby E{j and Gil and consider the master equation for the probability P( {E} , {G), t), which iswritten as_a_pc {E}, {c}, t) + ~ _a_. Fa P( {'E}, {cJ, t)at ii aGa=~ {W:(E~+c,E~,-c -+E~,E~,)p( E~+c E~I-c {C},t)aa' a a a a . .. a 'a '- ~,(E~,E~I-+E~+c E~,-c )P( ... E~,E~" .. {c},t)}a a a a 'a . a a+~ { G ~ --'>. --'>. --'>. --'>.---"-~~, w~ (G~ +g.... G~, - g4 -+G~ Gd, )P( {E} .. G +g-" G~, - g-" .,. t)a,a a a ' a a' a ' a ' a ' ,(17)where Fa is the conservative force (non-dissipative). The transition probability for the momentumchange is chosen to beG~ ~~ ~ ~ ~w~ (G- +g, G-+I - g -+ G~ , G~, )a a a a aEntropy of a system with convective velocity is given by (Martin et al (1972)),--'>.TdS = dE - pdp - u.dG .Hence-532-(18)(19)(20)A Note on Description of Hydrodynamic Fluctuationby a Master EquationWe define(21 )and(22)which appear in the expansion of the master equation (17) in the power ofg. These two quantitiesare related by(23)For small fluctuation of momentum, we findd ~ ~-<G- >=F. -}.;[fa«G-»-fa«G-,»]dt aa aa a' a aOn the other hand, from the Navier-Stokes equation2 2 ~I \7 fa ( < G- >) = Tl \72 UaaHenceTherefore by the relation (23), we find(24)(25)(26)(27)Thus all coefficients of the Kramers-Moyal expansion of the master equation up to the second de-rivatives are determined by the thermodynamic relations. So it would be interesting to discuss thefluctuation of hydrodynamic quantities in non-equilibrium conditions in the framework discussedabove.-533-

A note on Description of Hydrodyanmic Fluctuation by a Master Equation

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A note on Description of Hydrodyanmic Fluctuation by a Master Equation

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