Analysis of the Relativistic Brownian Motion in Momentum Space

Transcription

1 Brazilian Journal of Physics, vol. 36, no. 3A, eptember, Analysis of the Relativistic Brownian Motion in Momentum pace Kwok au Fa Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, , Maringá-PR, Brazil Received on 4 May, 006 We investigate the relativistic Brownian motion in the context of Fokker-Planck equation. Due to the multiplicative noise term of the corresponding relativistic Langevin equation many Fokker-Planck equations can be generated. Here, we only consider the to, tratonovich Hänggi-Klimontovich approaches. We analyze the behaviors of the second moment of momentum in terms of temperature. We show that the second moment increases with the temperature T for all three approaches. Also, we present differential equations for more complicated averages of the momentum. n a specific case, in the to approach, we can obtain an analytical solution of the temporal evolution of an average of the momentum. We present approximate solutions for the probability density for all three cases. Keywords: Relativistic Brownian motion; Langevin equation; Fokker-Planck equation. NTRODUCTON Nonrelativistic diffusion processes have been the subject of intense investigations in the last century. For instance, the well-known Brownian motion can be described by the usual nonrelativistic diffusion equation,. However, recently, several approaches have been used to model different kinds of anomalous diffusion processes 3, 4. n particular, the extension of the Brownian motion to the relativistic regime has been tackled by using several approaches 5 0. n the case of using the Langevin approach two types of Langevin equations have been used 7 0. One of them 7 the authors have postulated a constant diffusion coefficient the corresponding Langevin equation generates only one Fokker-Planck equation. Whereas, the other one 9 the authors have used the relativistic equation motions as a consequence the relativistic Langevin equation has a multiplicative noise term. n this last case, the one to one correspondence between the Fokker- Planck equation the Langevin equation can not be assured due to the different order of prescription in stochastic calculus. n fact, this Langevin equation can generate many different Fokker-Planck equations. n the paper 9 the authors have considered three different prescriptions which we follow them in our work. We analyze the behaviors of the second moment of momentum in terms of the temperature, for the stationary solutions. For the nonstationary solutions, the second moment is difficult to be obtained analytically. n this case, we investigate some more complicated averages of the momentum. We also present approximate solutions for the probability densities. These approximate solutions are used to investigate the second moments. p, t) t = ji // p,t) p i, ) where t is the time, p i are the relativistic momenta, j// i p,t) are the probability currents the indices are the abbreviations of tratonovich, to Hänggi- Klimontovich, respectively. For one-dimensional case the probability currents are given by j = νpρp,t) + D γp) ) γp)ρp,t), ) p j = νpρp,t) + D p γp)ρp,t)) 3) j = νpρp,t) + Dγp) p ρp,t)), 4) where γp) = + p /mc) ) /. For three-dimensional case the probability currents are given by j νp i = i ρp,t) + DL ) i k L ) T ) p jρp,t)) k, j 5). FOKKER-PLANCK EQUATON OF THE RELATVTC BROWNAN MOTON j i = νp i ρp,t) + D A ) i p jρp,t) ) j 6) n this section we recall the Fokker-Planck equations their stationary solutions obtained from the relativistic Brownian motion 9, 0, in the laboratory frame, which are given by j i = νp i ρp,t) + DA ) i j ρp,t)), 7) p j

2 778 Kwok au Fa where A L are the matrices associated with the momentum components 0. The stationary solutions of Eq.) are obtained by setting t = 0 j// i = 0. For one-dimensional case we have ρ p) = C 3 exp ), 3) ρ p) = ρ p) = ) C exp ) /4, 8) ) C exp ) / 9) where C 3, C 3 C 3 are the normalization factors.. NORMALZATON AND ECOND MOMENT FOR TATONARY OLUTON The above stationary solutions can be normalized analytically. The normalization of the stationary solutions permit us to analyze the behaviors of the second moments in terms of the temperature. For one-dimensional case we obtain the following normalized solutions: ρ p) = C exp ), 0) where C, C C are the normalization factors, = ν /D = mc /kt ), k is the Boltzmann constant T is the temperature. n the three-dimensional case the stationary solutions are given by ρ p) = ) C 3 exp ) 3, ) 4 ρ p) = ) πexp mc ) /4, 4) K 3 4 )K 4 ) ρ p) = exp ) / 5) mck 0 ) ) ρ p) = ) C 3 exp ) 3 ) ρ p) = ) exp mck ), 6) where K ν z) denotes the modified Hankel function. The corresponding second moments are given by = K 3 4 )K 4 ) { K 5 4 )K 3 4 ) K 4 )K } 4 ), 7) = K ) K 0 ) 8) = K ) K ). 9) n Fig. we show the second moment p, in one dimension,. We see that all three cases decrease with. This means that they increase with the temper-

3 Brazilian Journal of Physics, vol. 36, no. 3A, eptember, ature, just as in the classical case. Moreover, in the Hänggi- Klimontovich approach the second moment has the highest value among them, whereas in the classical case has the lowest value. For large values of all three cases converge to the classical one. = 4πmc)3 C 3 K ), 4) <p / > 0 M = K4 ) K ). 5) We note that the relations ) 5) have been obtained in 7. n Fig. we show the second moment p, in three dimensions,. All three cases also decrease with, just as in one-dimensional cases. For large value of the three cases converge to the classical one. We note that in the to approach the second moment has close values to those of the classical case, except for small. FG. : Plots of the second moments p,,,m, for one-dimensional stationary processes. The symbols,,, M correspond to the Hänggi-Klimontovich, tratonovich, to Maxwell distributions. n the case of three-dimensional processes we can also obtain the analytical normalization factors which are given by <p / > 00 0 M C3 = mc)3 π K 5 4 )K 3 4 ) K 4 )K, 4 ) 0) { C3 = πmc) 3 π F 3, ;,,; + 4 ) 4 +ln F ;,; 4 n=0 } n n ψ + n) + n)γ + n) ) C3 = 4πmc)3 K ), ) where ψz) is the Psi function, pf q a,...,ap ; b,...,b q ; z is the generalized hypergeometric function Γz is the Gamma function. The corresponding second moments of the momentum are given by ) = C 3, 3) C FG. : Plots of the second moments p,,,m, for three-dimensional stationary processes. The symbols,,, M correspond to the Hänggi-Klimontovich, tratonovich, to Maxwell distributions. V. AVERAGE OF THE MOMENTUM FOR ONE-DMENONAL CAE The n-moments are important physical quantities for the analysis of the stochastic processes. However, they are difficult to be obtained for the relativistic Brownian motion. n the above section we have only obtained the second moments for the stationary solutions. n order to obtain some average for the whole processes, we have found more complicated averages of the momentum by using the differential equation ). For the average γ a, where a is a real number, we can combine to obtain, by choosing different values of a, the differential equations for the averages of momentum. For the tratonovich approach we have

4 780 Kwok au Fa d γ 3 = 3ν γ 3 γ γ 6) n the case of the to approach we have the following differential equations: d γ 5 = 5ν γ γ 5 + γ 3 + γ. 7) For = we can obtain the following differential equation: d γ 5 5 d γ ν γ 5 5ν γ 3 = 0. 8) d γ 3 d γ = ν γ + ν γ 9) = 3ν γ 3 + γ + γ + They can be combined to form the expression. 30) d γ 3 + 3ν γ 3 = 3 d γ + ν + ) γ + ν. 3) We note that for = we can obtain the analytical solution which is given by γ 3 3 γ = Aexp 3νt), 3) where A is an integration constant. We see that the above average gives a simple exponential decay in time. Even the solution 3) is just valid for =, it can be applied to a wide range of diffusion coefficient D due to the relation = ν /D. However, it = ) determines the value of temperature which is given by T = mc /k). For the Hanggi-Klimontovich approach we consider the following averages: d γ dγ γ = ν γ γ + 33) = ν + γ γ γ 34) These equations can be combined to form the expression d γ +ν γ + dγ νγ = ν. 35) Unfortunately, only in the to approach one can solve the differential equation for the averages. We can check the solution. 3) by using the stationary solution 5). Numerical calculation yields γ 3 3 γ = ) which is in excellent agreement with the analytical result 3) p/mc FG. 3: Plots of the variations of νt) j p/mc) the nondimensional variable p mc by using the approximate solution Eq.37). The solid lines correspond to the variations of νt), whereas j the dotted lines correspond to the variations of. The lower p/mc) figure corresponds to νt = 0.5 the upper figure corresponds to νt = 3.

5 Brazilian Journal of Physics, vol. 36, no. 3A, eptember, p/mc <p / > 0. FG. 4: Plots of the variations of νt) j p/mc) the nondimensional variable p mc by using the approximate solution Eq.38). The solid lines correspond to the variations of νt), whereas j the dotted lines correspond to the variations of. The lower p/mc) figure corresponds to νt = 0.5 the upper figure corresponds to νt = νt FG. 5: Plots of the variations of νt) j in function p/mc) of the nondimensional variable p mc by using the approximate solution Eq.39). The solid lines correspond to the variations of νt), j whereas the dotted lines correspond to the variations of p/mc). The lower figure corresponds to νt = 0.5 the upper figure corresponds to νt = 3. p/mc FG. 6: Plots of the second moments p,, the nondimensional time νt by using the approximate solutions 37), 38) 39), for = 5. solutions are given by ρ p,t) = A exp exp ) + p p ) ) + exp ν + p t ) + p m ν c t + p ) ) /4, 37) V. ONE-DMENONAL APPROXMATE OLUTON We note that Eqs. ), ), 3) 4) are difficult to be solved analytically. n this case, approximate solutions can be useful to the analysis of the behaviors of probability densities. Also, they can be used to check the numerical results. n order to obtain our approximate solutions we have guided by the asymptotic solutions for p p. The approximate ρ p,t) = A exp exp ) + p p ) ) + exp ν + p t ) + p m ν c t + p ) ) / 38)

6 78 Kwok au Fa ρ p,t) = A exp exp ) + p p ) ) + exp ν + p t m c ) + p m ν c t + p where A, A A are the normalization factors. n Figs. 3, 4 5 we show the variations of ), 39) // νt) j // p/mc) obtained from the tratonovich, to Hänggi- Klimontovich approaches, respectively. We note that the behaviors of these quantities are similar for these three approaches. Also, these quantities are close together, specially for small νt. This means that our approximate solutions are in good approximations to the exact solutions. Moreover, we have analyzed numerically that the variations of // j // p/mc) νt) can maintain close together for not large t, greater than 3 ν less than. For large νt the variation between these two quantities is very small for p mc <,, in general, they maintain very close together for p mc >. From these approximate solutions we can calculate the second moments. n Fig. 6 we show the second moment p for the to, tratonovich Hänggi-Klimontovich approaches, for = 5. All three cases have similar behaviors to that of the classical Brownian motion. We have also checked for = 0, all three cases approximate closely to the result of classical theory. As has been numerically demonstrated in 9, the probability densities in velocity space approach a common Gaussian shape for large. V. CONCLUON We have analyzed the relativistic Brownian motion in the context of Fokker-Planck equations. We have put forward some analytical results in terms of the averages of momentum. We have shown an interesting analytical result in the to approach, i.e., the average γ 3 3 γ has a simple exponential decay in time. Further, approximate solutions for the probability densities have also been given for one-dimensional cases. We have used our approximate solutions to study the p temporal behaviors of the second moments, // the results of these three approaches converge to the classical result for large. On the other h, it has been emphasized in 9 that the three prescriptions discussed in this work, only the stationary solution of the Hänggi-Klimontovich approach is consistent with the relativistic Maxwell distribution. However, this last result is not the only one obtained by using the Langevin equation. n fact, the relativistic Maxwell distribution can also be obtained by using a different Langevin equation as that used by Debbasch 7, 8. Therefore, in order to choose which of the above approaches is the correct one, we need to know detailed information about the microscopic structure of the system. Finally, we hope that these analyses may contribute to a broad investigation of the relativistic Brownian motion. N. G. van Kampen, tochastic processes in physics chemistry North-Holl, 004). H. Risken, The Fokker-Planck equation pringer, 984). 3 R. Metzler J. Klafter, The rom walks s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339: 000). 4 R. Metzler J. Klafter, J. Phys. A 37, R6 004). 5 R. Hakim, J. Math. Phys. 6, ). 6 U. Ben-Ya acov, Phys. Rev. D 3, 44 98). 7 F. Debbasch, K. Mallick J. P. Rivet, J. tat. Phys. 88, ). 8 F. Debbasch J. P. Rivet, J. tat. Phys. 90, ). 9 J. Dunkel P. Hänggi, Phys. Rev. E 7, ). 0 J. Dunkel P. Hänggi, Phys. Rev. E 7, ).