Fluctuation-Dissipation Theorems in an Expanding Universe C. Chevalier and F. Debbasch ERGA-LERMA Universit´e Paris VI UMR 8112 3 rue Galil´ee 94200 Ivry, France

21st April 2006 Abstract The recently constructed minimal extension of the special relativistic Ornstein-Uhlenbeck process to curved space-time cannot be used to model realistically diffusions on cosmological scales because it does not take into account the evolution of the thermodynamical state of the matter with cosmological time. We therefore introduce a new class of non minimal extensions characterized by time-dependent friction and noise coefficients. These new processes admit time-dependent J¨ uttner distributions as possible measures in momentum space; associated fluctuation-dissipation theorems are also proved and discussed.

1

Introduction

The special relativistic Ornstein-Uhlenbeck process was introduced in 1997 [3] to model the stochastic dynamics of a point particle diffusing in flat space-time and surrounded by an isotropic fluid characterized by a possibly non-uniform 4-velocity field U . This process describes the interaction of the diffusing particle with the surrounding fluid through two forces; the first of these is deterministic and represents the effective frictional force experienced by the particle; the special relativistic Ornstein-Uhlenbeck process characterizes this force by a single friction coefficient α. The other force is stochastic and, being proportional to a Gaussian white noise, is fully determined by another single coefficient D which fixes its amplitude. Of particular importance is the fact that α and D are, like their Galilean analogues, both point- and momentum-independent; they are thus simply two real constants. It has been proven [3] that, for uniform velocity fields U , the special relativistic Ornstein-Uhlenbeck process admits as invariant measure in p-space the J¨ uttner distribution of temperature T = D/αmkB . This quantity can be interpreted as the equilibrium temperature of the surrounding fluid and its expression on terms of α and D constitutes a fluctuation-dissipation theorem. This special relativistic process has been recently minimally extended to curved space-times [4]. This extension still describes the force experienced by a diffusing particle through two coefficients α and D and, as in the flat space-time process, these quantities are taken to be both point- and momentum independent. This general relativistic process has been used to study diffusion in simple 1

cosmological models. The simplest possible model supposes the space-time to be spatially homogeneous and isotropic and describes the matter content of the universe by a perfect fluid; this fluid is sometimes called the cosmological fluid. Particularly convenient coordinates on the space-times are then the so-called comoving coordinates [6]; the cosmological fluid is at rest in these coordinates and all quantities characterizing the model depend only on the time-coordinate t. Because of the assumed homogeneity and isotropy of the space-time, the metric is fully determined by a single field a(t) called the expansion factor and the thermodynamical state of the matter present in the universe is entirely determined by a temperature field T (t). It is a standard result of theoretical cosmology [6] that T (t) varies approximately as 1/a(t). The minimal general relativistic extension of the special relativistic OrnsteinUhlenbeck process has been used to study how a particle diffuses in an homogeneous and isotropic universe through its interaction with the cosmological fluid [4]. It has been proven that the diffusion admits no invariant measure in p-space. Approximate spatially uniform but time-dependent solutions of the transport equation have been found in the physically relevant situations where the characteristic relaxation time-scale 1/α is much smaller than the Hubble time H −1 (t) = a/a. ˙ These solutions coincide with the time-independent J¨ uttner distribution of temperature T = D/αmkB in the limit where a becomes timeindependent i.e. when the space-time becomes flat and the process degenerates into the special relativistic Ornstein-Uhlenbeck process. But, for a general a(t), these solutions do not coincide with any time-dependent J¨ uttner distribution and there is therefore no time-dependent temperature naturally associated with the diffusion. This represents however a serious physical limitation of the minimally extended process introduced in [4] because, as mentioned earlier, the thermodynamical state of the matter in the universe can, to a very good approximation, be characterized by a time-dependent temperature T (t). The aim of the present article is to remedy this situation by introducing new, non minimal general relativistic extensions of the special relativistic OrnsteinUhlenbeck process which do admit time-dependent J¨ uttner distributions as possible densities in spatially homogeneous and isotropic universes. The material is organized as follows. Section 2 reviews some basic results about the special relativistic process and the associated flat space-time fluctuation-dissipation theorem. Section 3 describes rapidly the minimal extension constructed in [4] and its application to cosmological diffusions. Section 4 presents the new, non minimal extensions and the associated curved space-time fluctuation-dissipation theorems. More precisely, the class of non minimal extensions considered in this manuscript is described in Section 4.1. All considered processes are still fully characterized by two coefficients α and D only, but these are now allowed to depend on the cosmological time t and on the adimensionalized energy γ of the diffusing particle. We prove in Section 4.2 that, given any expansion factor a(t) and temperature field T (t), a process in the class considered will admit the J¨ uttner distribution of temperature T (t) as possible density in momentum space if, and only if, a combination ∆(t, γ) of the two coefficients α(t, γ) and D(t, γ) satisfy a certain differential equation; this equation is thus a curved space-time fluctuation-dissipation theorem in differential form. The integral form of the theorem is obtained in Section 4.3 and is an exact litteral expression for ∆(t, γ). This expression is the sum of two terms. The first one is a time-dependent entire series in 1/γ. The second term involves the exponential integral function 2

Ei, taken at mc2 γ/kB T (t) (where m is the mass of the diffusing particle). Our results are finally summed up and discussed in Section 5, where we also mention possible interesting developements.

2

The special relativistic Ornstein-Uhlenbeck process

The special relativistic Ornstein-Uhlenbeck process models the stochastic dynamics of a point particle diffusing in flat space-time because of its interaction with an isotropic fluid characterized by a 4-velocity field U . Suppose U is uniform. The fluid then admits a global proper frame R. The stochastic differential equations defining the process read in this frame ([3]): p dt mγ(p) √ p dt + 2DdBt , dp = −α γ(p) dx

=

(1)

where x and p are the position and momentum of the particle, and Bt is the usual 3 dimensional Brownian motion. The Lorentz factor γ reads in terms of p: p (2) γ(p) = 1 + p2 /m2 c2 .

The coefficient α fixes the amplitude of the frictional force, while the coefficient D gives the amplitude of the noise. The forward Kolmogorov equation corresponding to (2) and obeyed by the phase-space distribution Π(t, x, p)1 of the diffusing particle is ([3]):     p p ∂t Π + ∂x · Π − α ∂p · Π = D∆p Π. (3) mγ(p) γ(p)

The process admits as an invariant measure in p-space the J¨ uttner equilibrium distribution   1 1 γ(p) ΠJ (p) = exp − (4) 4π Q2 K2 (1/Q2 ) Q2

where K2 is the second order modified Hankel function and Q2 m2 c2 = D/α. The distribution ΠJ is normalized to unity with respect to the measure d3 p. The factor Q2 is interpreted as kB T /mc2 , where T is the point-independent equilibrium temperature of the surrounding fluid. One thus has: α=

D mkB T

(5)

which is a bona fide special relativistic fluctuation-dissipation theorem. By Lorentz transforming all quantities involved in equations (2), one can derive the stochastic equations of motion of the diffusing particle in an arbitrary Lorentz frame [1]. One can then obtain out of these stochastic equations the forward Kolmogorov equation associated to the process in any Lorentz frame. Equations (2) and (3) therefore define the process in all Lorentz frames ([1]), which makes this formulation of the ROUP perfectly covariant. 1 This

density is defined with respect to the Lebesgue measure d3 xd3 p

3

A manifestly covariant treatment of the special relativistic Ornstein-Uhlenbeck process has been developed in [2] and has been used in [4] to construct a minimal extension of this process to curved space-time. The next Section reviews the main properties of this minimal extension.

3

3.1

A minimal extension of the special relativistic Ornstein-Uhlenbeck process to curved spacetime Definition of the process

A manifestly covariant formulation of relativistic diffusion is achieved by working off the mass-shell; the standard 6-D phase-space is then replaced by an extended 8-D phase-space S which is best chosen as part of the bundle cotangent to spacetime2 . Any atlas of the space-time can be used to construct an atlas of S and coordinates in the extended phase-space will be hereafter designated by (x, q); note that we hereby introduce the notation q which will be used in the rest of this article to designate covariant momenta. Statistics in the extended phase-space are described by a scalar distribution function F. Given an arbitrary coordinate system, the distribution F is related to the physical distribution function Π to be used in this system by an equation of the form: Z F(x, q)δ(q0 − mcγ(t, x, q))dq0 , (6) Π(t, x, q) = R

where √ the δ-distribution enforces a mass-shell restriction. Integrating Π against d3 q/ −detg√delivers the spatial density n(x, t), which can in turn be integrated against d3 x −detg. Further details over the measures against which Π can be integrated can be found in [4]. A curved space-time generalization of the special relativistic Ornstein-Uhlenbeck process is then fully defined by a forward Kolmogorov equation obeyed by the function F. There naturally exists an infinity of such possible generalizations; the process introduced in [4] is defined by the simplest possible one; it reads: ∂ (mc Fdµ F) Dµ (g µν (x)pν F) + ∂pµ   pµ pβ µ β + DK ρ ν ∂pρ ∂p F = 0, pα U α ν

(7)

where D stands for the covariant derivative with respect to position at momentum covariantly constant: Dµ = ∇µ + Γα µν pα

∂ ; ∂pν

(8)

here, the Γ’s are the usual Christoffel symbols of the space-time metric g. The deterministic force Fd is defined by Fd

µ

= −λνµ pν

g µν pµ pν pα pβ + λαβ 2 2 pµ m 2 c2 m c

(9)

2 Naturally, the choice of S also ensures that, at any point of space-time, the fibre of S contains the mass-shell. The reader is referred to [4] for a precise definition of S

4

and the friction tensor λ reads λµν = α

(mc)2 P µν (pµ U µ )2

(10)

where P is for the projector onto the orthogonal to U : Pµν = gµν − Uµ Uν ;

(11)

finally, the tensor K is defined in terms of P and U by: K µρβν = U µ U β P ρν − U µ U ν P ρβ + U ρ U ν P µβ − U ρ U β P µν .

(12)

The manifestly covariant forward Kolmogorov equation for F can be transcribed in each coordinate system of space-time into a covariant but not manifestly covariant transport equation for the physical distribution function Π. This transport equation for Π fully defines the process in this coordinate system.

3.2

Application: diffusion in a spatially flat FRW universe

We first recall the expression of the components of the space-time metric characterizing the spatially flat FRW universe in the usual comoving coordinates [6]: ds2

= c2 dt2 − a2 (t)(dx21 + dx22 + dx33 ) 2

2

2

(13)

2

= c dt − a (t)dx .

The evolution of the expansion factor a(t) is fixed via Einstein’s equation by the equation of state of the matter present in the universe and by suitable initial data. The matter follows the expansion and is at rest in the comoving frame. The components of its 4-velocity U in the comoving frame are therefore: U µ = (1, 0, 0, 0).

(14)

Let Π(t, x, q) be the comoving frame distribution function of a particle whose interaction with the matter present in the Universe is fixed by the stochastic process considered in Section 3.1. This function satisfies the following forward Kolmogorov equation [4]:     q q 1 ∂x . Π + ∂q . α Π + D a2 (t)∂q Π , (15) ∂t Π = 2 a (t) m γ(t, q) γ(t, q) where the Lorentz factor γ(t, q) is given by s γ(t, q) =

1+

q2

a2 (t)m2 c2

.

(16)

The corresponding stochastic differential equations read: dx

=

dq

=

1 q dt a2 (t) m γ(t, q) √ q dt + 2 D a(t)dBt . −α γ(t, q)

−

5

(17)

¯ q) by: Let us define the marginal Π(t, Z ¯ q) = Π(t, x, q)d3 x; Π(t,

(18)

x∈IR3

¯ satisfies the transport equation: the distribution Π   q ¯ 2 ¯ ¯ ∂t Π = +∂q . α Π + D a (t)∂q Π , γ(t, q)

(19)

Unlike its galilean and special relativistic counterparts, the process defined by (17) is not time-homogeneous. In particular, equation (19) does not generally admit time-independent solutions. Pertubative expressions for solutions of (19) have however been found [4] in the case where the characteristic expansion timescale is much larger than the characteristic relaxation time 1/α of the process. When a(t) = 1, i.e. in flat space-time, these solutions reduce to the J¨ uttner equilibrium distribution of temperature Tf lat = D/αmkB . But, for a general a(t), these solutions do not coincide with any time-dependent J¨ uttner distribution and there is therefore no time-dependent temperature naturally associated with the diffusion. This represents however a serious physical limitation of the minimally extended process introduced in [4] because the thermodynamical state of the matter in the universe can, to a very good approximation, be characterized by a time-dependent temperature T (t) [6]. This limitation comes from the fact that the process (17), as a minimal extension of the special relativistic Ornstein-Uhlenbeck process, uses constant friction and noise coefficients α and D to characterize the interaction of the diffusing particle with the surrounding fluid. This simple choice has however no physical justification if the thermodynamical state of the fluid surrounding the particle is not itself constant and, for example, can be characterized by a time dependent temperature T (t). More realistic models of diffusion in a FRW space-time have to take into account the time-dependence of the thermodynamical state of the fluid by allowing the coefficients α and D to depend on time. The question which then naturally arises is: are there non minimal extensions of the special relativistic Ornstein-Uhlenbeck process to spatially flat FRW space-times which do admit as possible densities J¨ uttner distributions associated to time-dependent temperatures? To this we now turn our attention.

4 4.1

Fluctuation-dissipation theorems in an expanding universe Non minimal extensions of the ROUP in spatially flat FRW space-times

We consider the following class of Ito processes: dx = dq =

q 1 dt a2 (t) m γ(t, q) p q dt + 2 D(t, γ) a(t)dBt , −αI (t, γ) γ(t, q) −

(20)

where (t, x) are comoving coordinates. In (20), the coefficients α and D are allowed to depend on the comoving time t and on the adimensionalized energy 6

γ of the particle. The physical distribution Π(t, x, q) associated with one of these processes obeys the following forward Kolmogorov equation:     q 1 q Π + ∂q . α(t, γ) Π + D(t, γ) a2 (t)∂q Π (21) ∂t Π = 2 ∂x . a (t) m γ(t, q) γ where γ(t, q) is given by equation (16) and α(t, γ) = αI (t, γ) +

∂γ D . m 2 c2

(22)

¯ of Π defined by (18) then obeys the transport equation: The marginal Π   ¯ = +∂q . α(t, γ) q Π ¯ . ¯ + D(t, γ) a2 (t)∂q Π (23) ∂t Π γ R ¯ q)d3 q = 1 is conserved in time Note that the normalization condition IR3 Π(t, by this equation.

4.2

Differential form of the fluctuation-dissipation theorems

The time-dependent J¨ uttner distribution ΠJ (t, q) of temperature T (t) reads: ΠJ (t, q) = N (t) exp (−β(t)γ(t, q)) ,

(24)

2

with β(t) = kBmTc(t) . The function N (t) is determined by the normalization of ΠJ (t, q). We have: Z ΠJ (t, q)d3 q = 1, (25) R3

which yields:

N (t) =

β(t) 1 . 4 π a3 (t)(m c)3 K2 (β(t))

(26)

We now impose the J¨ uttner distribution (24) to be solution of the forward Kolmogorov equation (23):   q 2 (27) ∂t ΠJ = ∂q . α(t, γ) ΠJ + D(t, γ)a (t)∂q ΠJ γ and we consider (27) as a constraint on the coefficients α(t, γ) and D(t, γ) which define the non minimal extensions we wish to construct. We start by computing the left hand side of equation (27), using (24) and (26). We get: ! N˙ ˙ − β∂t γ , − βγ (28) ∂t ΠJ = ΠJ N where a dot denotes the derivative of a function of t. We have from (26): N˙ β˙ a˙ K ′ (β) = −3 + − β˙ 2 , N a β K2 (β)

7

(29)

and: ∂t γ = −

1 q2 a˙ ; 2 γ a (t)(mc)2 a

(30)

by using (16), this can be rewritten as: 1 a˙ ∂t γ = − (γ 2 − 1) , γ a

(31)

Plugging (29) and (31) into (28) leads to:   γ2 − 1 ΠJ , ∂t ΠJ = A1 (t) + A2 (t)γ + A3 (t) γ with

(32)

a˙ β˙ K ′ (β) A1 (t) = −3 + − β˙ 2 , a β K2 (β)

(33)

˙ A2 (t) = −β,

(34)

and

a˙ A3 (t) = β . a We now define the function ∆ by: ∆(t, γ) = α(t, γ) −

(35)

β(t)D(t, γ) m 2 c2

(36)

and express the right-hand side ρ(t, γ) of (27) in terms of ∆; one finds ρ(t, γ) =

ΠJ ∆ ΠJ ∆ ∂q .q + q.∂q ∆ + q.∂q ΠJ γ γ γ 1 + ∆ ΠJ q.∂q . γ

(37)

Expression q.∂q ∆ can be rewritten in terms of γ by deriving (16) with respect to q: γ2 − 1 ∂γ ∆. (38) q.∂q ∆ = γ 2

2

Immediate calculations also yield: q.∂q ΠJ = − β γ γ−1 ΠJ and q.∂q γ1 = − γ γ−1 3 . One thus finds    γ2 − 1 3γ 1 . (39) ∂γ ∆ + ∆ ρ(t, γ) = ΠJ −β− γ2 γ2 − 1 γ Plugging (32) and (38) into (27), we finally get: ∂γ ∆ + f (t, γ) ∆ = g(t, γ),

(40)

1 3γ − β(t) − γ2 − 1 γ

(41)

with f (t, γ) = and

γ2 g(t, γ) = 2 γ −1

  γ2 − 1 A1 (t) + A2 (t)γ + A3 (t) . γ 8

(42)

The above calculation shows that the J¨ uttner distribution (24) is a solution of (21) if and only if the coefficients α(t, γ) and D(t, γ) which define the diffusion process and the equilibrium temperature T (t) satisfy relation (40). Equation (40) is therefore a fluctuation-dissipation theorem in differential form. In flat Minkowski space-time and for a constant temperature T , a˙ = β˙ = 0 and a = 1; this implies A1 = A2 = A3 = 0 and (40) degenerates into ∂γ ∆ + f (γ)∆ = 0 (note that f is then time-independent). This differential equation can easily be integrated explicitely and the only solution which does not diverge when γ approaches unity, i.e. when p approaches zero, is ∆ = 0; the flat space-time fluctuation-dissipation theorem (5) presented in Section 2 is then recovered by choosing α and D independent of t and γ; remark however that this simple choice is clearly not the only one, even in flat space-time. Relation (40) is an ordinary first order differential equation with t- and γdependent coefficients f and g. In any arbitrary spatially flat universe, the function f depends on t via the inverse adimensionalized temperature β characterizing the state of the fluid present in the universe, while the comoving time dependency of g is induced not only by β but also by the purely geometric 3 quantity a/a ˙ . Hence, the function ∆, obtained by integrating (40) in γ, will depend on t through the functions a(t) and β(t), which are chosen to describe the geometry and the matter content of the space-time. The above discussion highlights the fact that there exist two different timescales involved in this problem: the Hubble time a/a˙ = H −1 of the expansion and the characteristic timescale β/β˙ associated with the evolution of the thermodynamical state of the matter.

4.3

Integral form of the fluctuation-dissipation theorems

Integrating the homogeneous equation ∂γ ∆H (t, γ) + f (t, γ)∆H (t, γ) = 0 associated with (40) yields the general solution: 2 −3/2 β(t)γ ∆H e . C (t, γ) = C(t) γ(γ − 1)

(44)

Note that, for any non vanishing value of C, ∆H C is not defined for γ = 1 (i.e. p = 0) and diverges as γ → 1+ . Any solution of the inhomogeneous equation (40) can be written as the sum P of a function ∆H C and of an arbitrary particular solution ∆ of (40). We write: ∆P (t, γ) = ∆0 (t, γ) γ(γ 2 − 1)−3/2 eβ(t)γ

(45)

and insert this expression into (40); this leads to:
∂γ ∆0 = e−β(t)γ (γ 2 − 1)1/2 A1 (t)γ + A2 (t)γ 2 + A3 (t)(γ 2 − 1) ,

(46)

3 The ratio a/a ˙ can be indeed expressed in terms of the Ricci tensor R of the space-time through the relation: „ «2 “ ” ” 1“ a˙ = 2Rµν g µν − Pαβ g αµ g βν − R (43) a 6

with P defined in (11). Note that this operator has a very simple geometric interpretation; it is indeed the projector onto the 3-surfaces of space-time invariant by the SO(3) symmetry group tracing for the spatial homogeneity and isotropy of the cosmological model.

9

with A1 (t), A2 (t) and A3 (t) defined by (33), (34) and (35). The rest of this section is devoted to finding the exact litteral expression of a particular ∆0 satisfying (46). This can be accomplished by performing a series expansion of equation (46) in powers of 1/γ 2. To do so, we first rewrite (46) as:   1/2 3/2 ! A1 1 1 3 −βγ ∂γ ∆0 = γ e 1− 2 . (47) + A2 + A3 1 − 2 γ γ γ 1/2 3/2   and 1 − γ12 can be expanded in powers of 1/γ 2 The functions 1 − γ12 and the expressions converge for all values of γ 6= 1. The series expansion of (47) reads:

∞ 1 1 1 k −βγ X 3 −βγ 2 −βγ 2 2 − 1 ... 2 − k + 1 (−1) e ∂γ ∆0 = (A2 + A3 )γ e + A1 γ e + A1 k! γ 2k−2 k=1 !



∞ 1 1 3 3 X − 1 ... 12 − k + 1 (−1)k − 1 ... 23 − k + 1 (−1)k e−βγ + A2 2 2 + A3 2 2 k! k! γ 2k−3 k=1

(48)

Rewriting the sums and putting together the first terms of each sum, which can be integrated straighforwardly, yields: A1 −βγ 1 e ∂γ ∆0 =(A2 + A3 )γ 3 e−βγ + A1 γ 2 e−βγ − (3A3 + A2 )γe−βγ − 2 2 ∞ ∞ (49) X e−βγ X e−βγ + bk 2k + (ck + dk ) 2k+1 , γ γ k=1

k=0

where the coefficients bk , ck and dk are given by:      1 1 (−1)k+1 1 −1 − 2 ... −k bk = A1 (k + 1)! 2 2 2       1 1 1 (−1)k −1 − 2 ... −k ck = A2 (k + 2)! 2 2 2       (−1)k 3 3 3 dk = A3 −1 − 2 ... −k (k + 2)! 2 2 2

(50)

A solution ∆S0 of (49) then reads:

∞

∞

X X A1 1 I4 + bk Jk + (ck + dk )J˜k , ∆S0 = (A2 + A3 )I1 + A1 I2 − (3A3 + A2 )I3 − 2 2 k=0 k=1 (51) with Z γ ′ I1 (t, γ) = γ ′3 e−βγ dγ ′ (52) ∞ Z γ ′ I2 (t, γ) = γ ′2 e−βγ dγ ′ Z∞γ ′ I3 (t, γ) = γ ′ e−βγ dγ ′ Z∞γ ′ e−βγ dγ ′ , I4 (t, γ) = ∞

10

which give I1 (t, γ) I2 (t, γ)


e−β(t)γ 3 β (t)γ 3 + 3β 2 (t)γ 2 + 6β(t)γ + 6 4 β (t)
e−β(t)γ 2 = − 3 β (t)γ 2 + 2β(t)γ + 2 β (t) = −

I3 (t, γ)

= −

e−β(t)γ (β(t)γ + 1) β 2 (t)

I4 (t, γ)

= −

e−β(t)γ , β(t)

(53)

and Jk (t, γ) = J˜k (t, γ) =

Z

γ

∞ Z γ ∞

′

e−β(t)γ dγ ′ γ ′2k

(54)

′

e−β(t)γ dγ ′ . γ ′2k+1

(55)

Both above integrals can be expressed in the form of series expansions for γ 6= 1 ([5]): 2k−1 X vj (−β)2k−1 Jk = −e−βγ + Ei(−βγ), (56) 2k−j γ (2k − 1)! j=1

with

vj = and

(−β)j−1 , (2k − 1)(2k − 2)...(2k − j)

J˜k = −e−βγ with v˜j =

2k X j=1

v˜j γ 2k+1−j

+

(−β)2k Ei(−βγ), (2k)!

(−β)j−1 . 2k(2k − 1)...(2k + 1 − j)

(57)

(58)

(59)

In (56) and (58), Ei is the exponential integral function, defined by ([5]): Z ∞ −x′ e dx′ , x < 0. (60) Ei(x) = − ′ −x x The function ∆S0 is defined for all values of γ 6= 1; since the right-hand side of (46) tends to 0 as γ approaches unity, all solutions of (46) are actually well behaved and therefore defined at γ = 1. The solution of (46) which coincides with ∆S0 for all values of γ 6= 1 will from now on be simply designated by ∆0 . Let us characterize further the behaviour of ∆0 near γ = 1. A direct expansion of (46) in powers of x = (γ 2 − 1)1/2 leads to ˜ 0 = e−β(t) (A1 (t) + A2 (t)) x2 + O(x4 ), ∂x ∆

(61)

˜ 0 (t, x) = ∆0 (t, (x2 + 1)1/2 ). Integrating this expression yields: where ∆ ˜ 0 (t, x) = ∆0 (t, 1) + e−β(t) (A1 (t) + A2 (t)) x3 + O(x5 ) ∆ 3 11

(62)

or, in terms of γ: ∆0 (t, γ) = ∆0 (t, 1)+e−β(t)


3/2
5/2  (A1 (t) + A2 (t)) 2 γ −1 +O γ 2 − 1 . (63) 3

Now, for any solution ∆ of (40), one can find a function C such that, for all values of γ 6= 1: ∆C (t, γ) = (C(t) + ∆0 (t, γ)) γ(γ 2 − 1)−3/2 eβ(t)γ .

(64)

Physics dictates that ∆ itself, and not only ∆0 , remains finite as γ approaches unity (i.e. as p approaches zero). Plugging (63) into (64) leads to: ∆C (t, γ) = (C(t) + ∆0 (t, 1)) γ(γ 2 −1)−3/2 eβ(t) +

(A1 (t) + A2 (t)) +O 3

γ2 − 1





.

(65) Choosing C(t) = −∆0 (t, 1) clearly ensures that the corresponding ∆ remains finite for γ = 1. The function ∆−∆0 (t,1) is therefore the unique physically relevant solution of (40). As indicated at the end of the previous section, the time-dependence of any solution of (40) and, in particular, the time-dependence of ∆−∆0 (t,1) , is governed by β(t) and by a(t). Except at very early epochs, the inverse adimensionalized temperature β(t) of our universe is roughly proportional to the expansion factor a(t) [6]; in this approximation, the Hubble time H −1 = a/a˙ = β/β˙ is the only time-scale of the problem and the natural interesting quantity is then the adimensionalized product ∆∗ (t, γ) = H −1 (t)∆−∆0 (t,1) (t, γ). The dependence of ∆∗ on γ is qualitatively similar for all values of t (or β) and a typical result is presented in Figure 1. D*

Β=1 2

4

6

8

10

Γ

-100 -200 -300 -400 -500 -600 -700

Figure 1: Evolution of ∆∗ (t, γ) versus γ for β = 1. The fact that ∆∗ is always negative admits of a very simple physical interpretation. It means indeed that, for all (t, q) and for any arbitrarily given value of the friction coefficient α(t, γ(t, q)), the amplitude of the noise needed to ensure thermal ‘equilibrium’ at temperature T (t) has to be greater in an expanding universe than in flat space-time. On the contrary, in a contracting universe, the quantity H −1 = a/a˙ would be negative and a negative ∆∗ would then correspond to a positive ∆; for any arbitrarily given value of α, the amplitude of the noise needed to ensure thermal equilibrium would then be smaller than in flat space-time. 12

5

Conclusion

Let us start by briefly summarizing our results. The thermodynamical state of the matter content of our universe can be characterized by a time-dependent temperature T (t). Any stochastic process modeling the diffusion of a point particle interacting with this matter should therefore admit the time-dependent J¨ uttner distribution of temperature T (t) as possible measure in momentum space. This is not the case for the minimal curved space-time extension of the special relativistic Ornstein-Uhlenbeck process. We have therefore introduced a class of new, non-minimal extensions of this process which all admit, in an expanding homogeneous and isotropic universe, the J¨ uttner distribution of temperature T (t) as possible measure in momentum space. These extensions are characterized by a friction and a noise coefficients which both depend on the energy of the diffusing particle and on the comoving time t; the time-dependence of these coefficients reflects the time-dependence of the thermodynamical state of the matter surrounding the diffusing particle; let us also note that this timedependence can be rephrased as a dependence on the Ricci tensor of the spacetime manifold. The two coefficients and the temperature T (t) obey a single differential equation which can be considered as the differential form of a fluctuation-dissipation theorem for the considered process. We have integrated this equation litteraly and have thus obtained an exact integral form of the fluctuation-dissipation theorem. The work presented in this article is suceptible of various extensions. First, one should systematically consider various astrophysical or cosmological situations where the thermodynamical state of the matter present in space-time can be adequately modelled by the introduction of a temperature field and investigate if one can always construct stochastic processes in these space-times which admit the associated J¨ uttner distribution as possible measure in momentum space. For example, can one prove fluctuation-dissipation theorems for diffusions inside a relativistic star, or in a non-homogeneous universe? And, if that is indeed the case, on which field characterizing the geometry of the space-time do the corresponding processes depend? In particular, for which space-times and/or temperature fields is it enough to consider a dependence on the Ricci tensor, and not on the full Riemann (or curvature) tensor? In a different direction, one should investigate if a theorem similar to the Htheorem recently proved [7] for the minimal extension of the special relativistic Ornstein-Uhlenbeck process can be obtained for the non minimal extensions considered in the present article and, more generally, for other curved spacetime extensions verifying physically interesting flcutuation-dissipation theorems.

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