Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator

Physica A xx (xxxx) xxx–xxx

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Physica A journal homepage: www.elsevier.com/locate/physa

Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator Q1

Emilio Cortés ∗ , J.I. Jiménez-Aquino Departamento de Física, Universidad Autónoma Metropolitana—Iztapalapa, Apartado Postal 55-534, C.P. 09340, México, Distrito Federal, Mexico

highlights • Hamilton–Jacobi and Fokker–Planck equations are equivalent for harmonic oscillators. • Extremal action supplies the solution of both equations. • The problem in presence of a magnetic field is explicitly solved.

article

info

Article history: Received 21 March 2014 Received in revised form 21 May 2014 Available online xxxx Keywords: Feynman’s path integral formalism Hamilton–Jacobi equation Fokker–Planck equation Ordinary Brownian harmonic oscillator Charged Brownian harmonic oscillator

abstract Using Feynman’s path integral formalism applied to stochastic classical processes, we show the equivalence between the Hamilton–Jacobi (HJ) and Fokker–Planck (FP) equations, associated with an overdamped Brownian harmonic oscillator. In this case, the Langevin equation leads to a Gaussian Lagrangian function and then the path integration which defines the conditional probability density can be replaced by the extremal path. Due to this fact and following the classical dynamics formalism, we prove the strict equivalence between the HJ and FP equations. We do this first for an ordinary Brownian harmonic oscillator and then the proof is extended to an electrically charged Brownian particle under the action of force fields: magnetic field and additional time-dependent force fields. We observe that this extremal action principle allows us to derive in a straightforward way not only the HJ differential equation, but also its solution, the extremal action. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Feynman’s path integral formalism (FPIF) [1] has been a useful tool to describe a variety of systems in the nonequilibrium statistical mechanics. It continues to be a very useful method in the study of classical [2–5] and quantum Brownian motion [6,7]. Between the eighties and nineties, the formalism was used to characterize the noise-induced linear and nonlinear stochastic dynamics driven by Gaussian colored noises [8]. The path integral formalism and its connection with the Hamiltonian dynamics has also been explored and applied to other situations where the stochastic fluctuations play a fundamental role [9]. In particular, in Ref. [10] a Hamiltonian formalism was given for a second order Langevin equation where an extremal action is formally written in terms of the nonlinear Hamilton equations. Due to the nonlinearity the expression for the extremal action in general is not easy to evaluate explicitly. Other works that use the Hamiltonian dynamics focus mainly on the probability distributions in the steady state, which are the time-independent solutions of the FP equations. It is shown that in the weak-noise limit of the steady state distribution, the FP equation reduces to a Hamilton–Jacobi-like equation [9,11].

∗

Corresponding author. E-mail addresses: [email protected] (E. Cortés), [email protected] (J.I. Jiménez-Aquino).

http://dx.doi.org/10.1016/j.physa.2014.05.064 0378-4371/© 2014 Elsevier B.V. All rights reserved.

1

2 3 4 5 6 7 8 9 10 11

2

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

27

From quantum mechanics point of view, it would be worth to comment that the FPIF was also used to obtain explicitly the Schrödinger equation from the Hamilton–Jacobi equation [12]. However, to the best of knowledge the explicit equivalence between the HJ and FP equations has not been reported in the literature even for linear equations. Our main contribution in this work is to use also the FPIF to show the equivalence between the HJ and FP equations for an overdamped Brownian harmonic oscillator, which satisfies a Langevin equation with additive Gaussian white noise (thermal noise). The equivalence is first proven for an ordinary Brownian harmonic oscillator (OBHO) and then when the oscillator is electrically charged, under the action of force fields: magnetic field and time-dependent force fields [13–17]. The Lagrangian is Gaussian in both cases which allows us (as a necessary condition) to replace the sum over paths by the extremal path [1]. With the extremal action defined in a formal way we follow the Hamiltonian formalism to obtain both, the extremal action by solving the Hamilton equations and also the HJ equation as a partial differential equation for that function. Once the HJ equation has been obtained in both cases, we show that it is totally equivalent to the FP equation through the relation W (x, t |x0 ) = N (t ) exp(−S [¯x]/4λ), where W (x, t |x0 ) is the transition probability density (TPD) associated with the FP equation, S [¯x] is the extremal action associated with the extremal path x¯ , and N (t ) is the normalization factor [18,19]. It must be noticed that the extremal action S is calculated for all time t ≥ 0 and as a consequence the probability density W is also given for all time t ≥ 0. Therefore, the equivalence means that the HJ equation is obtained from the FP equation not in the asymptotic regime but for all time t ≥ 0; besides the former equation does not mean a small-noise limit of the latter. The explicit expression for the extremal action gives the immediate solution of the FP equation. As a matter of fact, this approach which yields the extremal action from the Hamilton equations, constitutes an alternative way to obtain the solution of the FP equation. This work is then outlined as follows: in Section 2, we study the formalism of the extremal action and its corresponding HJ equation for an overdamped OBHO in the one dimensional case. The necessary and sufficient condition for the equivalence between the HJ and FP equations is established. Once the extremal action is calculated, the well known solution of the FP equation is easily verified. In Section 3, we extend the formalism of Section 2 to the case of an overdamped charged Brownian harmonic oscillator (CBHO) in the presence of a constant magnetic field only. In this case, the equivalence between the HJ and FP equations is established by a similar condition as in the OBHO. Also the solution of the FP equation is immediately obtained. The problem of an overdamped CBHO in the presence of additional time-dependent force fields is studied in Section 4. Our concluding remarks are given in Section 5, and at the end of our work two Appendices are included for explicit calculations.

28

2. Extremal action and HJ equation for an OBHO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

29 30

31 32 33

34 35 36 37 38 39 40

41 42

43

We start from a stochastic differential equation (SDE) given by a Langevin equation for a free harmonic particle in one dimension in the non-inertial regime (overdamped approximation)

γ x˙ + ω2 x = ξ (t ),

(1)

where γ = α/m is the friction coefficient α per unit mass, ω2 = k/m the oscillator’s characteristic frequency, and ξ (t ) the noise per unit mass. Let us assume a Gaussian noise with zero first moment and delta correlation function

⟨ξ (t )ξ (t ′ )⟩ = 2λ δ(t − t ′ ),

(2)

where λ is the noise intensity which satisfies the fluctuation–dissipation relation λ = γ kB T /m, with kB the Boltzmann constant and T the temperature of the surrounding medium (the thermal bath). In order to calculate the extremal action of this problem, we follow Feynman’s functional formalism [1] for a stochastic process. Due to the fact that this stochastic differential equation is linear with additive noise, it turns out that noise probability distribution has a one-to-one correspondence with the distribution of the dynamical variable x(t ). Taking into account the Gaussian character of the noise, then we start from a Lagrangian function which represents the systematic part of the Langevin equation.1 L(˙x(t ), x(t )) = (γ x˙ + ω2 x)2 ,

(3)

and the action functional reads S [x(t )] =



t2

L(˙x(t ), x(t )) dt .

(4)

t1 44 45

46

47 48

The integral is evaluated along a path x(t ) with fixed end points (x1 , t1 ) and (x2 , t2 ). From this, the conditional probability density can be formally defined as a path integral W (x, t |x0 ) = N (t )



exp([−S [x(t )]/4λ]) D [x(t )],

(5)

D [x(t )] means a path differential and N (t ) is a normalization factor. According to Feynman and Hibbs [1], if L(˙x(t ), x(t )) is Gaussian, which means a function up to second degree in its variables, then the sum over the paths can

Q2 in which

1 Here the Lagrangian function has a different definition from that of classical dynamics.

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

3

be substituted by an extremal path. This is equivalent to establish an extremal action principle to define the conditional probability density W (x, t |x0 ) = N (t ) exp([−S [¯x(t )]/4λ]),

(6)

where S [¯x(t )] stands for the action evaluated along the extremal path, or just the extremal action. We now apply an extremal principle to the action functional (4) to obtain the corresponding Hamiltonian function associated with the extremal path. First we write the expression for the conjugate momentum p=

∂L = 2γ (γ x˙ + ω2 x), ∂ x˙

(7)

from which we can obtain the Hamiltonian function p

H = x˙ p − L =

2

4γ 2

− ax p,

p ∂H = − ax, ∂p 2γ 2

p˙ = −

∂H = ap. ∂x

(8)

p

,

p(t ) = p0 e ,

8aγ 2

(13)

We now have to write the initial value p0 in terms of x0 , x, and t which are the initial and final values of the variable, and the final time value. In order to get this relation we integrate first Eq. (9) to obtain x(t ′ ) = c1 e

at ′

x˙ (t ) = ac1 e ′

+ c2 e

at ′

−at ′

− ac2 e

,

−at ′

.

p0 = 4aγ c1 .

15 16

19

21 22 23

24

25 26

(14)

27

(15)

28

Combining these equations with Eqs. (9) and (12) we get 2

14

20

(12)

(e2at − 1).

13

18

where t is the time variable and t is used for the time value at the end point of the path. Using Eqs. (11), (12), and the integration of L over the interval [0, t ], as in Eq. (4), we obtain the action S=

9

17

′

p20

7

12

being p the solution of Eq. (10) which is given by at ′

6

(10)

(11)

′

5

11

From Eqs. (3) and (9) we can write the extremal Lagrangian as 4γ 2

4

(9)

2.1. Extremal action for an OBHO

L=

3

10

The extremal action and its corresponding HJ equation come from two routes: the extremal action comes from the solution of the Hamilton equations (9) and (10), and the HJ equation, which is a partial differential equation for the extremal action, is obtained from the Hamiltonian function equation (8). In other words, we can obtain independently the partial differential equation for the extremal action and its solution without solving the equation. We will do this for the OBHO.

2

2

8

where a = ω2 /γ = k/α . The Hamilton equations are the differential equations for the extremal path and are given by x˙ =

1

29

(16)

Applying the boundary conditions to Eq. (14) we have

30 31

c1 + c2 = x0 ,

(17)

32

c1 eat + c2 e−at = x,

(18)

33

from which we can write p0 as p0 = 4aγ 2

34

( x − x0 e ) . (eat − e−at ) −at

(19)

Inserting this expression into Eq. (13) we get finally the extremal action S (x, t ) = 2 aγ 2

( x − x0 e ) . (1 − e−2at )

35

36

−at 2

(20)

37

4

1

2 3

4

5 6

7

8

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2.2. HJ equation for an OBHO By considering the extremal action equation (4) as a function of the coordinate and time at the upper limit of integration, the variation of this action can be expressed as

δS =



∂L δx ∂ x˙

t2



t2



+ t1

t1

 ∂L d ∂L − δ x dt . ∂ x dt ∂ x˙

(21)

Since all the possible paths satisfy the Lagrange equations, the integral term is zero. Then it is shown [20] the following relation ∂S = p. (22) ∂x Because S = S (x, t ) we thus have

∂S ∂S ∂S = + x˙ = + x˙ p, dt ∂t ∂x ∂t and from Eq. (4) we get dS

9

10

(23)

dS

11

12

13

14

15

= L. dt From the last two expressions and using also the Hamiltonian function (8), we arrive to the HJ equation ∂S + H = 0. ∂t

(25)

According to Eqs. (8) and (22) the HJ equation can be written as

∂S 1 + ∂t 4γ 2



∂S ∂x

2

 − ax

∂S ∂x



= 0.

(26)

16

It can be verified by direct substitution that the extremal action given by Eq. (20) is the solution of this HJ equation.

17

2.3. Equivalence between HJ and FP equations

18 19

20

21 22 23

24

25 26

The FP equation for the transition probability density W (x, t |x0 ) associated with the Langevin equation (1) is given by [21–23]

∂W ∂ xW λ ∂ 2W =a + 2 , (27) ∂t ∂x γ ∂ x2 being D = λ/γ 2 = kB T /α the Einstein’s diffusion coefficient. We now proceed to show the equivalence between both the HJ equation (26) and FP equation (27). First we derive the HJ equation starting from the FP equation (27) using Eq. (6). It is not difficult to show after some straightforward algebra that

∂S 1 + ∂t 4γ 2

28 29 30

32 33 34 35 36

37

∂S ∂x

2

∂S − ax ∂x 



− 4λ

˙ N N

+

1 ∂ 2S 4γ 2 ∂ x2



− a = 0.

(28)

N

Φ (t ) =

a

, (29) 4γ 1 − e−2at and then the last parenthesis in Eq. (28) is just a function of time. Due to the fact that the stochastic process given by Eq. (1) is stationary, its associated FP equation (27) as well as Eq. (28) cannot contain a purely time-dependent term. Under these conditions, the whole parenthesis should be equal to zero, that is 2

˙ N 31



From expression (20) we see that S is quadratic in x and therefore its Laplacian (∂ 2 Sx /∂ x2 ) will be just a function of time t, that is (∂ 2 Sx /∂ x2 ) ≡ Φ (t ). This will always be the case when the Lagrangian function is Gaussian. Thus we have 1

27

(24)

+

1 ∂ 2S 4γ 2 ∂ x2

 −a =0

(30)

and therefore Eq. (28) reduces to the HJ equation (26). This condition allows to determine the normalization factor which is then given by N (t ) = C0 (1 − e−2at )−1/2 ,

(31)

where the constant C0 can√be calculated from the normalization condition for the transition probability density (6). In this √ case C0 = 1/ 2π D/a = β k/2π being β = 1/kB T and the normalization factor becomes N (t ) =



βk 2π (1 − e−2at )

1/2

.

(32)

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

5

This factor along with the extremal action (20) determines the transition probability density W (x, t |x0 ) such that

  β k (x − e−at x0 )2 W (x, t |x0 ) = N (t ) exp − , 2 (1 − e−2a8t )

1

(33)

2

which is precisely the solution of the FP equation (27). In the opposite direction, we start from the HJ equation (26) to obtain the FP equation (27). This is explicitly shown in Appendix A, and the condition to arrive to the FP equation (27) is exactly the same as that given by the last parenthesis of Eq. (28), showing then the complete equivalence between both equations. Respect to the structure of these two equations, a remark can be made. The FP equation is linear in the differential operators and it contains a Laplacian term, whereas the HJ equation is not linear in its differential operators, but it does not contain a Laplacian term.

3 4 5 6 7 8

3. Extremal action and HJ equation for a CBHO in a magnetic field

9

Consider now an electrically charged Brownian harmonic oscillator with charge q in the presence of a constant magnetic field. We also consider the overdamped approximation of its associated Langevin equation which can be written in vector form as q v × B = ξ(t ), (34) γ r˙ + ω2 r − mc where again the noise term per unit mass ξ(t ) satisfies the property of a Gaussian white noise with zero mean value and correlation function

⟨ξi (t )ξj (t )⟩ = 2λ δij δ(t − t ), ′

′

i, j = x, y, z ,

(35)

being λ = γ kB T /m the noise intensity. If the magnetic field points along the z-axis, then the Langevin equation can be written in terms of its components as

10 11 12

13

14

Q3

15 16 17 18

γ x˙ + ω x − by˙ = ξx (t ),

(36)

19

γ y˙ + ω y + bx˙ = ξy (t ),

(37)

20

γ z˙ + ω2 z = ξz (t ),

(38)

21

2

2

being b = qB/mc. It can be observed in this set of equations that, due to the presence of the magnetic field the Langevin equations (36) and (37) in the (x, y) plane are two coupled stochastic differential equations and independent of Eq. (38) along the z-axis. This latter is the same as that given by Eq. (1), so that we will be able to separate the z component from the (x, y) plane. 3.1. HJ equation in a magnetic field

(39)

The conjugate momenta become

∂L = 2[(γ 2 + b2 )˙x + ω2 (γ x + by)], ∂ x˙ ∂L py = = 2[(γ 2 + b2 )˙y + ω2 (γ y − bx)], ∂ y˙ ∂L = 2γ [γ z˙ + ω2 z ]. pz = ∂ z˙ px =

L=

+

p2y

4(γ + 2

b2

)

+

p2z 4γ 2

,

c¯ 2

(p2x + p2y ) +

γ

¯ )px + (¯ay − bx ¯ )py ] − azpz , − [(¯ax + by

28

29

(41)

32

(42)

33

34

35

36

(44)

Using Eqs. (40)–(42), this Hamiltonian function can be written in term of coordinates and momenta as H =

27

31

which does not depend explicitly on the coordinates. On the other hand, the Hamiltonian function is defined by

p2z 4 2

25

(40)

(43)

H = x˙ px + y˙ py + z˙ pz − L.

24

30

After some algebraic steps, the Lagrangian function (39) can be written in terms of the momenta as p2x

23

26

In this section we first focus on the formalism of the HJ equation. According to the Gaussian character of the noise, the Lagrangian of the system becomes L = (γ x˙ − by˙ + ω2 x)2 + (γ y˙ + bx˙ + ω2 y)2 + (γ z˙ + ω2 z )2 .

22

37 38

(45)

39

6

1 2

where we have defined the parameters a¯ = a/(1 + C 2 ), b¯ = a¯ b/γ = aC /(1 + C 2 ), c¯ = 1/2γ 2 (1 + C 2 ), and C = b/γ = qB/α c. We can also define the Lagrangian and Hamiltonian functions in the (x, y) plane only as

5 6

11

12

13 14

15

16

17

18 19 20

(p2x + p2y ),

(46)

c¯ 2 ¯ )px + (¯ay − bx ¯ )py ]. (px + p2y ) − [(¯ax + by (47) 2 We now construct the HJ equation in the (x, y) plane in a similar way as done in the preceding section. In this case we also have px =

∂ Sxy , ∂x

py =

∂ Sxy , ∂y

(48)

and the HJ equation is again

∂ Sxy + Hxy = 0, ∂t

9 10

2

Hxy =

7

8

c¯

Lxy =

3

4

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

(49)

or c¯ ∂ Sxy + ∂t 2



∂ Sxy ∂x

2

 +

∂ Sxy ∂y

2 

    ∂ Sxy ∂ Sxy ¯ ¯ − (¯ax + by) − (¯ay − bx) = 0. ∂x ∂y

(50)

3.2. Equivalence between HJ and FP equations in a magnetic field The FP equation for the transition probability density Wxy (x, t |x0 ), associated with the coupled Langevin equations (36) and (37) is shown to be [14–17]

∂ Wxy = ∇x · (Ax Wxy ) +  λ ∇x2 Wxy , ∂t where the position vector x = (x, y),  λ = λ/γ 2 (1 + C 2 ) = D/(1 + C 2 ), and the matrix A reads   a¯ b¯ A= . −b¯ a¯

(51)

(52)

The FP equation associated with the Langevin equation (38) is exactly the same as that given in Eq. (27) but along the z-axis. The solution of the FP equation (51) in terms of the extremal action is given by Wxy (x, t |x0 ) = Nxy (t ) exp([−Sxy [¯x(t )]/4λ]),

(53)

25

where x¯ (t ) is the extremal trajectory and Sxy must be the solution of Eq. (50). As shown in Appendix B, the FP equation (51) is derived from the HJ equation (50) if the condition given by Eq. (108) holds. In similar way as shown in Section 2, this condition determines the normalization factor Nxy if the extremal action is known. In the opposite direction, the HJ equation (50) can be derived from the FP equation (51) and the condition for the fulfillment of this statement is also the same as that given by Eq. (108).

26

3.3. Extremal action in a magnetic field

21 22 23 24

27 28

29

30

31

32

33

From the Hamiltonian equation (47) we derive the Hamilton equations for the extremal path in the (x, y) plane. These equations then are

∂ Hxy ¯ , = c¯ px − a¯ x − by ∂ px ∂ Hxy ¯ , y˙ = = c¯ py − a¯ y + bx ∂ py x˙ =

∂ Hxy ¯ y, = a¯ px − bp ∂x ∂ Hxy ¯ x + a¯ py . p˙ y = − = bp ∂y p˙ x = −

(55) (56) (57)

In vector form we just have dx

34

(54)

dt

= c¯ p − A x,

(58)

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

7

where the matrix A is same as Eq. (52). For the momentum vector p = (px , py ) we get dp dt

1

= Bp,

(59)

and

2

3

 B=

 −b¯ . a¯

a¯ b¯

(60)

As we can see the matrix A is the transpose of B and therefore A + B = 2a¯ I, with I the unit matrix. The solution of Eq. (59) is readily obtained as p(t ) = e

p0 ,

Bt

(61)

which substituted into Eq. (58) yields

= c¯ eBt p0 − A x. dt The solution of this equation reads

(62)

p0 =

c¯ (e2a¯ t − 1)

eAt (x − e−At x0 ).

(63)

|p|2 = (p2x + p2y ) = e

(64)

W=

(p2x0 + p2y0 ) = e

|p0 |2 . −At

13

15

x0 and notice that the matrix

16

b¯ . 0

17



0

−b¯

R(t ) = eWt =

(66)



¯ cos bt ¯ − sin bt

¯ sin bt ¯ cos bt



,

2a¯

2a¯ ea¯ t

eAt X =

− 1) c¯ (e2a¯ t − 1) Due to the fact that RĎ (t )R(t ) = I, then e2a¯ t

(67)

|p0 |2 = pĎ0 p0 =

2

4a¯ e

2a¯ t

c¯ 2 (e2a¯ t − 1)2

R(t )X.

′

(68)

22

23

|x − e−At x0 |2 ,

(69)

24

25

2

4a¯ e

2a¯ t

c¯ 2 (e2a¯ t − 1)2

|x − e−At x0 |2 ,

(70)

′ and from Eq. (46) we have that Lxy = 2c¯ |p|2 = 2c¯ |p0 |2 e2a¯ t . The integration of Lxy over t ′ defines the extremal action Sxy , which is finally given by

Sxy = 2a γ 2

20

21

and we conclude that

|p|2 = e2a¯ t

18

19

which has the property RĎ (t ) = R−1 (t ), i.e., the transposed matrix is its inverse. Thus, Eq. (64) can now be written as c¯ (

11

(65)

It can be shown that the exponential eAt = ea¯ t eWt = ea¯ t R(t ), where R(t ) is a rotation matrix given by [24]

p0 =

9

14

2a¯ t ′

The norm of p0 can be calculated in the following way. We first define the vector X = x − e A = a¯ I + W, where W is the antisymmetric matrix



7

12

From the solution (61) it is easy to see that 2a¯ t ′

6

10

c¯

(e2a¯ t − 1)e−At p0 . 2a¯ Solving for p0 in terms of the initial and final values of the coordinates, as well as the final time we get 2a¯

5

8

dx

x(t ) = e−At x0 +

4

−A t

26

27 28

2

|x − e x0 | . (1 − e−2a¯ t )

(71)

It can be shown that this extremal action is the solution of the HJ equation (50). Given this extremal action, the solution of the FP equation (51) then reads



Wxy (x, t |x0 ) = Nxy (t ) exp −

β k |x − e−At x0 | 2 (1 − e−2a¯ t )

 2

,

(72)

where the normalization factor is given by Eq. (109). By the way, this solution has also been obtained in Ref. [17], where a transformation of the Langevin equation (34) by means of a time-dependent rotation matrix was used. The solution along

29

30 31

32

33

8

1

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

the z-axis is the same as Eq. (33), i.e.,



2

Wz (z , t |z0 ) = Nz (t ) exp −

 β k (z − e−at z0 ) , 2 (1 − e−2at )

(73)

4

and Nz (t ) is the same as Eq. (32). Finally, the solution of the FP equation associated with a charged Brownian harmonic oscillator in a constant magnetic field is given by the product of Eqs. (72) and (73), that is W (r, t |r0 ) = Wxy (x, t |x0 )Wz (z , t |z0 ).

5

4. The presence of additional force fields

3

6 7

Consider now the presence of additional and arbitrary time-dependent force fields Fe (t ) in the Langevin dynamics of the above charged Brownian harmonic oscillator. In this case, the set of stochastic equations are given by

8

γ x˙ + ω2 x − by˙ − aex (t ) = ξx (t ),

(74)

9

γ y˙ + ω y + bx˙ − aey (t ) = ξy (t ),

(75)

10 11 12 13

2

γ z˙ + ω z − aez (t ) = ξz (t ), (76) where the acceleration is ae (t ) = Fe (t )/m. The problem can also be formulated in the (x, y) plane independently of the z-axis. To solve the problem, we introduce a new variable R = r − ⟨r⟩, such that R = (X , Y , Z ). In this case it can be shown that the variables (X , YZ ) satisfy the following Langevin equations 2

14

γ X˙ + ω2 X − bY˙ = ξx (t ),

(77)

15

γ Y˙ + ω Y + bX˙ = ξy (t ),

(78)

16 17

2

γ Z˙ + ω2 Z = ξz (t ). However, the variables ⟨r⟩ = (⟨x⟩, ⟨y⟩, ⟨z ⟩) satisfy the deterministic equations

18

γ

19

γ

20

21 22 23

24

25

26

27 28 29 30

31 32

33

34 35 36

37

d⟨x⟩ dt d⟨y⟩ dt d⟨z ⟩

+ ω2 ⟨x⟩ − b + ω2 ⟨y⟩ + b

d⟨y⟩ dt d⟨x⟩ dt

(79)

= aex (t ),

(80)

= aey (t ),

(81)

+ ω2 ⟨z ⟩ = aez (t ). (82) dt As we can see, Eqs. (77)–(79) have the same mathematical structure than those given in Eqs. (36)–(38). Also Eqs. (77) and (78) are independent of Eq. (79). Following exactly the same algebraic steps given in Section 4, it can be shown that the extremal action in the (X , Y ) variables reads γ

SXY = 2aγ 2

|R¯ − e−At R¯ 0 |2 , (1 − e−2a¯ t )

(83)

¯ = (X , Y ). The same happens with the Z variable, giving as a result where R SZ = 2aγ 2

(Z − e−at Z0 )2 . (1 − e−2at )

(84)

¯ = x − ⟨x⟩ and Z = z − ⟨z ⟩, where ⟨x⟩ and ⟨z ⟩ are To return to the original variables, we just substitute the variables R respectively the deterministic solutions of Eqs. (80)–(82). According to Eqs. (80) and (81), the solution for ⟨x⟩ reads ⟨x⟩ = e−At ⟨x(0)⟩ + e−At ¯f(t ),  t −A s where ¯f(t ) = (1/ω2 ) e A a¯ e (s) ds is a two dimensional vector. In this case it is shown that

(85)

0

¯ −e R

−At

¯ 0 = x − e−At [x0 + ¯f(t )], R

(86)

and therefore the extremal action in the original variables will be

|x − e−At [x0 + ¯f(t )]|2 . (1 − e−2a¯ t ) In the case of the ⟨z ⟩ variable we get the solution Sxy = 2aγ 2

⟨z ⟩ = e−at ⟨z (0)⟩ + e−at fz (t ), t where fz (t ) = γ −1 0 eas aez (s) ds and then the extremal action becomes Sz = 2aγ 2

[z − e−at (z0 + fz (t ))]2 . (1 − e−2a¯ t )

(87)

(88)

(89)

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

9

On the other hand, the FP equation associated with the coupled Langevin equations (74), (75), and that associated with Eq. (76) for the respective transition probability densities W xy (x, t |x0 ) and W z (z , t |z0 ) are given by

1 2

∂ W xy + ae · ∇x W xy = ∇x · (Ax W xy ) +  λ ∇x2 W xy , ∂t

(90)

3

∂W z ∂ xWx λ ∂ 2 Wx ∂W z + aez =a + 2 . ∂t ∂x ∂x γ ∂ x2

(91)

4

Finally, the solution of the FP equation associated with the Langevin equations (74)–(76) is given by W (r, t |r0 ) = W xy (x, t |x0 )W z (z , t |z0 ), where

6

  Sxy βk exp , ¯ − 2 a t 2π(1 − e ) 4λ

W xy (x, t |x0 ) =

(92)

and

5

7

8

W z (z , t |z0 ) =

1/2

βk



 exp

2π (1 − e−2at )



Sz 4λ

.

(93)

5. Concluding remarks

10

By means of Feynman’s path integral formalism applied to stochastic classical dynamics, we have shown the strict equivalence between the HJ and FP equations for both OBHO and CBHO in the overdamped approximation of their associated Langevin equations. In both cases, the equivalence is established by the necessary and sufficient conditions given in Eqs. (99) and (108) of Appendices A and B. The extremal action for both harmonic oscillators are solutions of their respective HJ equation, and they have been calculated without the necessity to solve these partial differential equations. Once we have obtained the extremal actions, the solutions of the FP equations come immediately. In spite of the equivalence of both HJ and FP equations, we would like to highlight the difference in the mathematical structure between them. We notice that the FP equation is linear in the differential operators and it contains a Laplacian term, whereas the related HJ equation is not linear in its differential operators, but it does not contain a Laplacian term. We observe that the Laplacian of S function does not depend on the coordinates due to the Gaussian nature of the Lagrangian function, and so it does not appear in the HJ equation. On the other hand, the Laplacian of W function does depend on the coordinates because of the exponential relation W = N (t ) exp(−S /4λ). On the other hand, the problem of the CBHO in the presence of additional time-dependent force fields is immediately solved in terms of the variables (X , Y , Z ) which satisfy the Langevin equations similar as those given when only the magnetic field is present. Lastly, the study of the Langevin equation including the inertial term is in progress. We believe that those works related to noise-induced scape rate over a potential barrier driven by Gaussian white and colored noises using the FPIF [8], can in principle be extended to the case of a Brownian particle in the presence of a constant magnetic field. However we want to stress here that higher order potential functions, beyond the quadratic, lead to nonlinear Langevin equations and therefore to non-Gaussian Lagrangians; so in these cases the extremal paths will not represent rigorously the sum over paths, and approximate solutions can be obtained in the small noise limit where the path integral can be evaluated by the method of steepest descents [8]. We also think that the present contribution, allows to explore the solution of the FP equation associated with other coupled Langevin equations as the one recently studied in Ref. [25] in other context. Appendix A. FP equation coming from HJ equation for an OBHO

∂W = ∂t

N

−

1 ∂S 4λ ∂ t



W,

∂W W ∂S =− , ∂x 4λ ∂ x   2  ∂ 2W W ∂ 2S 1 ∂S =− − . ∂ x2 4λ ∂ x2 4λ ∂ x

∂W ∂W − ax ∂t ∂x

 −

λ γ2



∂ 2W ∂ x2

 −W

˙ N N

+

1 4γ 2



∂ 2S ∂ x2



12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

34

(94)

35

(95)

36

(96)

37

We now substitute the corresponding expressions into Eq. (26) to obtain



11

33

Here we will show how to obtain the FP equation (27) from the HJ equation (26), starting from Eq. (6). In this case

˙ N

9

= 0.

38

(97)

39

10

1

2

3 4

E. Cortés, J.I. Jiménez-Aquino / Physica A xx (xxxx) xxx–xxx

But −ax(∂ W /∂ x) = −a(∂ xW /∂ x) + aW , and then Eq. (97) is rewritten as

    ˙  2   ∂W ∂ xW λ ∂ 2W N 1 ∂ S −a − 2 −W + − a = 0. ∂t ∂x γ ∂ x2 N 4γ 2 ∂ x2

In a similar way as commented in Section 2.2, because of the stationarity of the Langevin equation (1), the last term of Eq. (98) must be equal to zero, that is N˙

5

N

+



1 4γ 2

∂ 2S ∂ x2



− a = 0,

(99)

6

and then the FP equation (27) is obtained from the HJ equation (26).

7

Appendix B. FP equation coming from HJ equation for a CBHO in a magnetic field

8

9

10

11

Starting from Eq. (53) we show that

∂ Wxy = ∂t

∂ 2 Wxy ∂ x2

13

∂ 2 Wxy ∂ y2

15

16

17

18

19

20

21 22 23 24

26

27

−

Nxy

1 ∂ Sxy 4λ ∂ t



Wxy ,

(100)

∂ Sxy , ∂x ∂ Sxy , ∂y     Wxy ∂ 2 Sxy 1 ∂ Sxy 2 =− − , 4λ ∂ x2 4λ ∂x     Wxy ∂ 2 Sxy 1 ∂ Sxy 2 =− − . 4λ ∂ y2 4λ ∂y

(101) (102)

(103)

(104)

Upon substitution of the corresponding expressions into the HJ equation (50) we arrive to

     2  2 ∂ Wxy ¯ ) ∂ Wxy − (¯ay − bx ¯ ) ∂ Wxy − 2λ¯c ∂ Wxy + ∂ Wxy − (¯ax + by ∂t ∂x ∂y ∂ x2 ∂ y2 ˙  2  Nxy c¯ ∂ Sxy ∂ 2 Sxy − Wxy + + = 0. Nxy 2 ∂ x2 ∂ y2 ¯ ) − (¯ax + by



∂ Wxy ∂x



¯ ) − (¯ay − bx



∂ Wxy ∂y



= ∇x · (A xWxy ) − 2a¯ Wxy .

(106)

Hence, taking into account that 2λ¯c = γ 2 (1λ+C 2 ) =  λ, Eq. (105) reduces to

∂ Wxy − ∇x · (A xWxy ) −  λ∇x2 Wxy − Wxy ∂t

˙ Nxy Nxy

 c¯ + ∇x2 Sxy − 2a¯ = 0. 2

(107)

It must be noticed that the last term in the parenthesis of Eq. (107) is given in the two dimensions, and it is quite similar as that given in Eq. (98) for one dimension. As we can see, the extremal action Sxy of Eq. (71) is also quadratic in x, and then its Laplacian is only function of time. The Langevin equations (36)–(38) are also stationary processes and therefore the last term of Eq. (107) must satisfy the condition

Nxy

c¯ + ∇x2 Sxy − 2a¯ = 0, 2

(108)

which also determines the normalization factor Nxy (t ), that is Nxy (t ) =

βk 2π (1 − e−2a¯ t )

.

28

Hence, Eq. (107) reduces finally to the FP equation (51) as expected.

29

References

30 31

(105)

The second and third terms can be shown to satisfy

N˙ xy 25

˙ Nxy

∂ Wxy Wxy =− ∂x 4λ ∂ Wxy Wxy =− ∂y 4λ

12

14

(98)

[1] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. [2] H.S. Wio, J. Phys. A: Math. Theor. 46 (2013) 115005.

(109)

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11

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