Spectral decomposition approach to macroscopic parameters of Fokker-Planck flows: Part 2
SSpectral decomposition approach to macroscopic parameters ofFokker-Planck flows: Part 2
Igor A. TanskiMoscow, [email protected]
In this paper we proceed with investigation of connections between Fokker -Planck equation and continuum mechanics. We base upon expressions from our work[2], based upon the spectral decomposition of Fokker - Planck equation solution. In thisdecomposition we preserve only terms with the smallest degrees of damping. We find,that macroscopic parameters of Fokker-Planck flows, obtained in this way, satisfy the setof conservation laws of classic hydrodynamics. The expression for stresses (30) containsadditional term - this term is negligible in big times limit. We proved also, that the veloc-ities field alone satisfy Burgers equation without mass forces - but with some additionalterm. This term is also negligible in big times limit. For the zero degree theory, consid-ered in [1], there are no additional terms. But this theory is valid only for the potentialvelocities field, fully deductible from density - the potential is proportional to density log-arithm. In this theory we can not specify initial conditions for velocities independentlyfrom density. Taking in account of the next degree terms could partly solve this problem,but result in some loss of exactness.
Keywords
Fokker-Planck equation, continuum mechanicsThis paper is the second part of our work [1]. The aim of these papers is to investigate connectionsbetween Fokker - Planck equation and continuum mechanics. We follow the method, proposed in [4].Namely, in spectral decomposition of solution we preserve only terms with the smallest degree of damping.In the first part we considered only zero degree terms. Here we discuss results of the next degree termsconsideration. In the course of calculations we use results of our work [3].We know from [1], that for zero degree terms satisfy the set of classic hydrodynamics equations forisothermal compressible fluid with friction mass force, proportional to velocity. Velocities alone satisfy inthis case the Burgers equation without mass forces. But all this is true only for the potential velocities field,fully deductible from density - the potential is proportional to density logarithm. This is, of course, veryspecial class of flows. In particular, we can not specify initial conditions for velocities independently fromdensity.2-We expect, that the taking in account of he next degree terms could solve this problem.All references to formulae [1] we shall write according to following format: (P1-
1. Density
To get the first approximation of Fokker - Planck equation solution we keep terms with p i = r = + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) exp ØŒŒº - t (cid:230)(cid:231)Ł a + k a j = S w j (cid:246)(cid:247)ł øœœß · (1) · A w w w (cid:230)Ł k pa (cid:246)ł (cid:230)Ł k a (cid:246)ł (cid:230)Ł i w a (cid:246)ł j = j = P exp( i w j x j ) exp ØŒº - k a (cid:230)Ł w j a (cid:246)ł øœß d w d w d w .It is useful to express r i as a derivative of some potential f i to simplify handling terms with negativedegrees of w j in following expressions (see, for example (7)). Besides that we write e - a t factor explicitlyto have clear insight of damping velocities of different terms in following expressions. We write r = e - a t ¶ f ¶ x ; (2)where f = + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) exp ØŒº - t k a j = S w j øœß · (3) · A w w w (cid:230)Ł k pa (cid:246)ł (cid:230)Ł k a (cid:246)ł (cid:230)Ł a (cid:246)ł j = j = P exp( i w j x j ) exp ØŒº - k a (cid:230)Ł w j a (cid:246)ł øœß d w d w d w .We see, that potential f satisfy diffusion equation ¶ f ¶ t = k a (cid:230)Ł ¶ f ¶ x + ¶ f ¶ y + ¶ f ¶ z (cid:246)ł , (4)and density satisfy diffusion with damping equation ¶ r ¶ t + ar = k a (cid:230)Ł ¶ r ¶ x + ¶ r ¶ y + ¶ r ¶ z (cid:246)ł . (5)Tw o another functions r , r are defined in the similar to (2) way by potentials f and f . Thesepotentials satisfy the same equation (4), densities satisfy (5). Let us now denote the previously considered3-density (P1-8) as r . Then full density r for first degree approximation can be expressed as a sum r = r + r + r + r = r + e - a t ¶ f k ¶ x k . (6)Here and in the following repeated indices are understood to be summed (Einstein’s summation con-vention).
2. Velocities
We hav e following expressions for velocities u k = i r + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) exp ØŒŒº - t (cid:230)(cid:231)Ł a + k a j = S w j (cid:246)(cid:247)ł øœœß · (7) · A w w w (cid:230)Ł k pa (cid:246)ł ØŒº - k a w k a - ad k w øœß (cid:230)Ł k a (cid:246)ł (cid:230)Ł i w a (cid:246)ł j = j = P exp( i w j x j ) exp ØŒº - k a (cid:230)Ł w j a (cid:246)ł øœß d w d w d w .where r is defined by (6). Using this definition, we have u k = e - a t r (cid:230)Ł - k a ¶ f ¶ x ¶ x k + ad k f (cid:246)ł . (8)Recall expression (P1-11) for velocities u k = r (cid:230)Ł - k a ¶ r ¶ x k (cid:246)ł (9)and have u k = u k + u k + u k + u k , (10)or u k = r (cid:230)Ł - k a ¶ r ¶ x k + a e - a t f k (cid:246)ł . (11)
3. The continuity equation
As we mentioned in [1], though we dropped all fast damping terms, (6) and (11) are still derivedfrom exact solution of Fokker - Planck equation. Therefore they must satisfy certain conservation laws.4-Let us check the continuity equation. ¶ ( r u k ) ¶ x k = (cid:230)Ł ¶ ( r u k ) ¶ x k + ¶ ( r u k ) ¶ x k + ¶ ( r u k ) ¶ x k + ¶ ( r u k ) ¶ x k (cid:246)ł = (12) = (cid:230)Ł - k a ¶ r ¶ x k - k a ¶ r ¶ x k - k a ¶ r ¶ x k - k a ¶ r ¶ x k + a e - a t ¶ f k ¶ x k (cid:246)ł = - ¶ r ¶ t - ¶ r ¶ t - ¶ r ¶ t - ¶ r ¶ t = - ¶ r ¶ t .and we get the continuity equation ¶ r ¶ t + ¶ ( r u k ) ¶ x k =
0. (13)We can eliminate velocities and write the continuity equation in another form ¶ r ¶ t - k a ¶ r ¶ x k ¶ x k + a e - a t ¶ f k ¶ x k =
0. (14)
4. Expressions for r derivatives We can express derivatives of r k in terms of velocities u k . For example, expression (P1-15) readsnow ¶ r ¶ x j = - a k r u j , (15)and similarly ¶ r ¶ x j = - a k (cid:230)Ł r u j - a e - a t d k f (cid:246)ł . (16)Sum (15) and three expressions (16) and get ¶ r ¶ x j = - a k (cid:230)Ł r u j - a e - a t f j (cid:246)ł . (17)in full agreement with (11).This means, that the field (cid:230)Ł r u i - a e - a t f i (cid:246)ł posses potential. This potential is equal to (cid:230)Ł - k a r (cid:246)ł .We can also express second derivatives of r k in terms of velocities u k and their derivatives. Forexample, differentiation of (15) gives5- ¶ r ¶ x i ¶ x j = - a k ¶ r ¶ x i u j - a k r ¶ u j ¶ x i . (18)Pick expression for ¶ r ¶ x j from (17) and insert it to (18) ¶ r ¶ x i ¶ x j = (cid:230)Ł a k (cid:246)ł (cid:230)Ł r u i - a e - a t f i (cid:246)ł u j - a k r ¶ u j ¶ x i . (19)Similarly differentiation of (16) gives ¶ r ¶ x i ¶ x j = - a k ¶ r ¶ x i u j - a k (cid:230)Ł r ¶ u j ¶ x i - a e - a t d k ¶ f ¶ x i (cid:246)ł = (20) = (cid:230)Ł a k (cid:246)ł (cid:230)Ł r u i - a e - a t f i (cid:246)ł u j - a k (cid:230)Ł r ¶ u j ¶ x i - a e - a t d k ¶ f ¶ x i (cid:246)ł .Sum (19) and three expressions (20) and get ¶ r ¶ x i ¶ x j = (cid:230)Ł a k (cid:246)ł (cid:230)Ł r u i - a e - a t f i (cid:246)ł u j - a k (cid:230)Ł r ¶ u j ¶ x i - a e - a t ¶ f j ¶ x i (cid:246)ł . (21)Alternate (21) on the indices i , j (cid:230)Ł r ¶ u j ¶ x i - a e - a t ¶ f j ¶ x i (cid:246)ł - (cid:230)Ł r ¶ u i ¶ x j - a e - a t ¶ f i ¶ x j (cid:246)ł + a k a e - a t ( f i u j - f j u i ) =
0. (22)We shall use this identity (see (41)). ¶ r ¶ x i ¶ x j = (cid:230)Ł a k (cid:246)ł (cid:230)Ł r u i u j - a e - a t f i u j - a e - a t f j u i (cid:246)ł - a k ØŒº r (cid:230)Ł ¶ u j ¶ x i + ¶ u i ¶ x j (cid:246)ł - a e - a t (cid:230)Ł ¶ f j ¶ x i + ¶ f i ¶ x j (cid:246)ł øœß . (23)Another interesting identity follows from (23) and (14) k a k = S (cid:230)Ł ¶ r ¶ x k ¶ x k - a e - a t ¶ f k ¶ x k (cid:246)ł = ¶ r ¶ t = k = S (cid:230)Ł a k (cid:230)Ł r u k - a e - a t f k (cid:246)ł u k - r ¶ u k ¶ x k (cid:246)ł . (24)
5. Current of momentum tensor and stresses
The expression for current of momentum tensor is (see [2] and [3]):6- J kl = + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) + ¥ - ¥ (cid:242) exp ØŒŒº - t (cid:230)(cid:231)Ł a + k a j = S w j (cid:246)(cid:247)ł øœœß · (25) · A w w w (cid:230)Ł k pa (cid:246)ł (cid:230)(cid:231)Ł ØŒº - k a w k a - ad k w øœß ØŒº - k a w l a - ad l w øœß - d kl ØŒº k a + a d l w øœß (cid:246)(cid:247)ł ·· (cid:230)Ł k a (cid:246)ł (cid:230)Ł i w a (cid:246)ł j = j = P exp( i w j x j ) exp ØŒº - k a (cid:230)Ł w j a (cid:246)ł øœß d w d w d w .We see after, that simplification (cid:230)Ł - k a w k a - ad k w (cid:246)ł (cid:230)Ł - k a w l a - ad l w (cid:246)ł - d kl (cid:230)Ł k a + a d l w (cid:246)ł = (26) = (cid:230)Ł k a (cid:246)ł w k w l + d k w k a w l + d l w k a w k - d kl k a .cancel all negative degree terms. J kl = e - a t ØŒº (cid:230)Ł k a (cid:246)ł ¶ f ¶ x ¶ x k ¶ x l - d k k a ¶ f ¶ x l - d l k a ¶ f ¶ x k + d kl k a ¶ f ¶ x øœß . (27)Recall (P1-14) J ij = (cid:230)Ł k a (cid:246)ł ¶ r ¶ x i ¶ x j + k a r d ij . (28)Sum (28) and tree expressions (27) J kl = J kl + J kl + J kl + J kl = (cid:230)Ł k a (cid:246)ł ¶ r ¶ x k ¶ x l - k a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł + d kl k a r = (29) = (cid:230)Ł r u k - a e - a t f k (cid:246)ł u l - (cid:230)Ł k a (cid:246)ł (cid:230)Ł r ¶ u l ¶ x k - a e - a t ¶ f l ¶ x k (cid:246)ł - k a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł + d kl k a r == غ r u k u l - a e - a t ( f k u l + f k u l ) øß - k a ØŒº r (cid:230)Ł ¶ u k ¶ x l + ¶ u l ¶ x k (cid:246)ł - a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł øœß - k a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł + d kl k a r == غ r u k u l - a e - a t ( f k u l + f k u l ) øß - k a ØŒº r (cid:230)Ł ¶ u k ¶ x l + ¶ u l ¶ x k (cid:246)ł + a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł øœß + d kl k a r .Tensor of stresses components are equal to7- s kl = r u k u l - J kl = a e - a t f k u l + k a (cid:230)Ł r ¶ u l ¶ x k + a e - a t ¶ f k ¶ x l (cid:246)ł - d kl k a r = (30) = a e - a t ( f k u l + f l u k ) + k a ØŒº r (cid:230)Ł ¶ u k ¶ x l + ¶ u l ¶ x k (cid:246)ł + a e - a t (cid:230)Ł ¶ f k ¶ x l + ¶ f l ¶ x k (cid:246)ł øœß - d kl k a r .Recall, that in [1] we get the state equation of compressible viscous fluid with kinematic viscosityequal to n = k a . We see, that additional term is present in (30) - the first term. This term is proportional to e - a t and therefore it is relatively small for big t .
6. Equations of movement
Now we consider another conservation law - equation of movement. The equation is: ¶ ( r u i ) ¶ t + ¶ ( r u i u j ) ¶ x j - ¶ s ij ¶ x j + ar u i =
0. (31)We make following substitutions in (31) (all expressions are known from the previous text) : r u k = - k a ¶ r ¶ x k + a e - a t f k . (32) r u k u l - s kl = J kl . (33)The result is: ¶¶ t (cid:230)Ł - k a ¶ r ¶ x i + a e - a t f i (cid:246)ł + (34) + ¶¶ x j ØŒº (cid:230)Ł r u i u j - a e - a t ( f i u j + f j u i ) (cid:246)ł - k a ØŒº r (cid:230)Ł ¶ u i ¶ x j + ¶ u j ¶ x i (cid:246)ł + a e - a t (cid:230)Ł ¶ f i ¶ x j + ¶ f j ¶ x i (cid:246)ł øœß + d ij k a r øœß ++ ar u i = ¶ r ¶ t = - k = S ¶ ( r u k ) ¶ x k ; (35) ¶ f i ¶ t = k a (cid:230)Ł ¶ f i ¶ x + ¶ f i ¶ y + ¶ f i ¶ z (cid:246)ł ; (36)8- ¶¶ x j (cid:230)Ł d ij k a r (cid:246)ł = k a ¶ r ¶ x i = - a (cid:230)Ł r u i - a e - a t f i (cid:246)ł (37)and get k a ¶ ( r u j ) ¶ x i ¶ x j - a e - a t f i + a e - a t ¶ f i ¶ x j x j + (38) + ¶¶ x j (cid:236)(cid:237)(cid:238) (cid:230)Ł r u i u j - a e - a t ( f i u j + f j u i ) (cid:246)ł - k a ØŒº r (cid:230)Ł ¶ u i ¶ x j + ¶ u j ¶ x i (cid:246)ł + a e - a t (cid:230)Ł ¶ f i ¶ x j + ¶ f j ¶ x i (cid:246)ł øœß (cid:252)(cid:253)(cid:254) - a (cid:230)Ł r u i - a e - a t f i (cid:246)ł ++ ar u i = ¶¶ x j (cid:236)(cid:237)(cid:238) k a ¶ ( r u j ) ¶ x i + a e - a t ¶ f i ¶ x j + (cid:230)Ł r u i u j - a e - a t ( f i u j + f j u i ) (cid:246)ł - (39) - k a ØŒº r (cid:230)Ł ¶ u i ¶ x j + ¶ u j ¶ x i (cid:246)ł + a e - a t (cid:230)Ł ¶ f i ¶ x j + ¶ f j ¶ x i (cid:246)ł øœß (cid:252)(cid:253)(cid:254) = ¶¶ x j (cid:236)(cid:237)(cid:238) - (cid:230)Ł r u i - a e - a t f i (cid:246)ł u j + k a r ¶ u j ¶ x i + a e - a t ¶ f i ¶ x j + (cid:230)Ł r u i u j - a e - a t ( f i u j + f j u i ) (cid:246)ł - (40) - k a ØŒº r (cid:230)Ł ¶ u i ¶ x j + ¶ u j ¶ x i (cid:246)ł + a e - a t (cid:230)Ł ¶ f i ¶ x j + ¶ f j ¶ x i (cid:246)ł øœß (cid:252)(cid:253)(cid:254) = ¶¶ x j (cid:236)(cid:237)(cid:238) (cid:230)Ł a e - a t ( f i u j - f j u i ) (cid:246)ł + k a ØŒº r (cid:230)Ł ¶ u i ¶ x j - ¶ u j ¶ x i (cid:246)ł + a e - a t (cid:230)Ł ¶ f i ¶ x j - ¶ f j ¶ x i (cid:246)ł øœß (cid:252)(cid:253)(cid:254) =
0. (41)This equality is true because of identity (22).In this way we checked, that variables r and v i satisfy the set of conservation laws of classic hydro-dynamics. They are not quite the set of equations of classic hydrodynamics of isothermal compressiblefluid, because expression for stresses (30) contains additional term. This term is negligible in big timeslimit.
7. Burgers equation ¶ u i ¶ t + u j ¶ u i ¶ x j - r ¶ s ij ¶ x j + a u i =
0. (42)We take expression for stresses in asymmetric form (see 30)) s ij = a e - a t f j u i + k a r ¶ u i ¶ x j + k a e - a t ¶ f j ¶ x i - d ij k a r . (43)Perform substitution ¶ u i ¶ t + u j ¶ u i ¶ x j - r ¶¶ x j (cid:230)Ł a e - a t f j u i + k a r ¶ u i ¶ x j + k a e - a t ¶ f j ¶ x i - d ij k a r (cid:246)ł + a u i = ¶ u i ¶ t + u j ¶ u i ¶ x j - r e - a t ¶¶ x j (cid:230)Ł af j u i + k a ¶ f j ¶ x i (cid:246)ł - (45) - r k a ¶ r ¶ x j ¶ u i ¶ x j - k a ¶ u i ¶ x j ¶ x j + r k a ¶ r ¶ x i + a u i = ¶ r ¶ x j = - a k (cid:230)Ł r u j - a e - a t f j (cid:246)ł (46)once again: ¶ u i ¶ t + u j ¶ u i ¶ x j - r e - a t ¶¶ x j (cid:230)Ł af j u i + k a ¶ f j ¶ x i (cid:246)ł + (47) + r (cid:230)Ł r u j - a e - a t f j (cid:246)ł ¶ u i ¶ x j - k a ¶ u i ¶ x j ¶ x j - r a (cid:230)Ł r u i - a e - a t f i (cid:246)ł + a u i = ¶ u i ¶ t + u j ¶ u i ¶ x j - k a ¶ u i ¶ x j ¶ x j - r e - a t ¶¶ x j (cid:230)Ł af j u i + k a ¶ f j ¶ x i (cid:246)ł - (48) - r a e - a t f j ¶ u i ¶ x j + r a e - a t f i = ¶ u i ¶ t + u j ¶ u i ¶ x j - k a ¶ u i ¶ x j ¶ x j + (49) + r e - a t ØŒº a f i - ¶¶ x j (cid:230)Ł af j u i + k a ¶ f j ¶ x i (cid:246)ł - af j ¶ u i ¶ x j øœß = u i only. Similarly to [1], this equation strongly resemblesBurgers equation. To get more usual form of Burgers equation we could perform substitution t ¢ = t and n ¢ = k a , but we omit this calculation.We see from (49), that velocities v i satisfy the Burgers equation with some additional term. This term(the last term in (49)) is negligible in big times limit. Moreover, only this last term depend on r variable. Itis inversely proportional to r and directly proportional to f and its derivatives. DISCUSSION