Probabilistic Derivation of Spatiotemporal Correlation Functions in the

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Probabilistic Derivation of Spatiotemporal Correlation Functions in the Hydrodynamic Limit Giovanni Ciccotti, Sara Bonella, Mauro Ferrario, and Carlo Pierleoni J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b10066 • Publication Date (Web): 31 Dec 2015 Downloaded from http://pubs.acs.org on January 19, 2016

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The Journal of Physical Chemistry

Probabilistic Derivation of Spatiotemporal Correlation Functions in the Hydrodynamic Limit Giovanni Ciccotti,∗,†,‡ Sara Bonella,∗,¶ Mauro Ferrario,∗,§ and Carlo Pierleoni∗,k †Dipartimento di Fisica, University of Roma “La Sapienza”, Rome, Italy 1

‡School of Physics, University College Dublin (UCD), Dublin, Ireland ¶CECAM, Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland §Dipartimento di Scienze Fisiche, Informatiche e Matematiche, University of Modena and Reggio Emilia, Modena, Italy kDipartimento di Scienze Fisiche e Chimiche, University of L’Aquila, L’Aquila, Italy E-mail: [email protected]; [email protected]; [email protected]; [email protected] Phone: +39 06 49914378

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Abstract

2

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In this paper we use probability theory to prove, in suitable conditions, the equiv-

4

alence of equilibrium time correlation functions of microscopic density fields with the

5

time correlation functions of local macroscopic density fields evolved by hydrodynamics

6

in (approximate) phenomenological continuum theories of matter. We further discuss

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a useful and rigorous numerical algorithm, derived from this framework, to compute

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macroscopic space- and time-dependent behaviors (such as the hydrodynamical one)

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via molecular dynamics simulations.

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Introduction

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In Statistical Mechanics the time dependent behavior of many particle systems is described

12

via the definition of appropriate time correlation functions, which are related to the transport

13

properties of the system by linear response theory. 1–3 These correlation functions are defined

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formally as statistical averages of appropriate phase space observables but their explicit eval-

15

uation at the microscopic level is, in general, impossible except via numerical simulations.

16

In this paper, we shall consider in particular the time dependent correlation functions of

17

the densities of conserved quantities. When the density fields to be correlated are far apart

18

in space and time, a macroscopic approach has been developed to gain at least qualitative

19

insight in their behavior. In these conditions, in fact, their behavior can be identified with

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that of “hydrodynamical” macroscopic fields. 4,5 Under the assumption of local thermody-

21

namic equilibrium, the dynamics of these fields is known thanks to the conservation laws

22

and the constitutive relations.

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This hydrodynamic description was built first for the mass (or particle) density and the

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velocity fields (as functions of space and time) together with the temperature and pressure

25

fields. The extension to other cases, e.g. chemical kinetics, has been also considered. 6 In

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the works of Kadanoff and Martin, 5 and of Mountain 7 the time evolution of the hydrody-

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namic fields has been connected with the equilibrium time correlation functions via linear 2


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response theory. An alternative, direct derivation provided in the literature and in reference

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textbooks 8 has instead left a conceptual gap and some possible confusion.

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In this paper we show how to obtain the connection between macroscopic fields and

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time correlation functions using a precise probabilistic definition that does not require linear

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response theory. We then discuss the specific application of the procedure to the density-

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density correlation function of simple fluids to obtain the expression of the van Hove function 4

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in the hydrodynamic limit. The paper is organised as follows. We start by recalling some

35

necessary basic results of probability theory. We then prove the equivalence of the time cor-

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relation functions of microscopic observables over the equilibrium phase space ensemble with

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the time correlation functions of the local macroscopic fields introduced in the continuum

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theories of matter. In the last section we discuss, referring to applications already appeared

39

in literature, how this idea can generate a useful algorithm to get the macroscopic behavior

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of density fields from microscopic atomistic simulation. Finally, we draw some conclusions.

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The probability framework

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Let us begin by recalling a few definitions and the basic rules of probability theory that

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we need. 9 For the sake of simplicity we shall restrict this preliminary discussion to the case

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of two random variables. The more complex setting addressed in the next two sections is

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conceptually equivalent to the discussion below.

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47

48

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Let us consider the pair of stochastic variables W = (X, Y ) distributed continuously in the region D = (a, b) × (c, d) according to their joint probability density fW (x, y). We will RR indicate joint averages via the usual angular bracket notation h · · · iW = D · · · fW (x, y)dxdy. The definition of the related marginal probability density for the variable X is

fX (x) =

50

Z

d

fW (x, y)dy .

(1)

c

from which, whenever legitimate, the conditional probability density of Y given that X = x 3



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can also be defined as

fc (y|x) ≡

fW (x, y) fX (x)

(2)

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(symmetrically, the marginal for Y and the conditional for X given that Y=y can be also

53

defined, but they are not necessary here).

54

The ‘conditional’ average of Y can now be defined as

hy|xiY = 55

Z

d

yfc (y|x)dy = c

Z

d

y c

fW (x, y) dy fX (x)

(3)

which can then be averaged over X to give Z

b

hy|xiY fX (x)dx Z b Z d fW (x, y) = dy fX (x)dx y fX (x) a c ZZ yfW (x, y)dxdy = hyiW =

h hy|xiY iX =

a

(4)

D

56

If instead we consider the average of the product between x and hy|xiY we get h x hy|xiY iX = =

57

58

Z

b

ZaZ

x hy|xiY fX (x)dx D

xy fW (x, y)dxdy = hxyiW

(5)

i.e the correlation of X and Y , where for simplicity we have redefined, as usual, the random variables so that hxiW = hyiW = 0.

59

Density fields and microscopic description

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Let us now consider an ensemble of systems of N particles described by the time-independent

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Hamiltonian HN (Γ) where Γ ≡ {R1 , . . . , RN , P1 , . . . , PN } is a phase space point. For future 4


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63

64


convenience, we also define Γt ≡ {R1 (t|Γ), . . . , RN (t|Γ), P1 (t|Γ), . . . , PN (t|Γ)}, the space point evolved at time t via Hamilton’s equations with initial conditions Γ ≡ Γt=0 . The macroscopic density field ρ(r, t) is defined as

ρ(r, t) = PN

Z

dΓf (Γ, t)ˆ ̺(r|Γ)

(6)

65

where ̺ˆ(r|Γ) =

66

parametric function of the phase space point Γ), and f (Γ, t) is the probability density. Note

67

that the time dependence in f (Γ, t) arises because, in a hydrodynamic process, the initial

68

condition for the probability density is not the stationary equilibrium. Thus, even though

69

the Hamiltonian, and therefore the equilibrium statistical ensemble, are not time dependent,

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the macroscopic field evolves in time due to the relaxation of the non equilibrium initial

71

condition f (Γ, 0). Writing Hamilton’s equations as Γ˙ = iLΓ = {HN , Γ}, where L is the

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Liouville operator of the dynamics governed by HN (Γ), and introducing the time evolution

73

operator S(t, 0) = eiLt such that Γt (Γ) = S(t, 0)Γ, the evolution of the probability density,

74

solution of the Liouville equation, can be written formally as 10

i=1

δ(r − Ri ) is the microscopic density (a field in the position r and a

f (Γ, t) = S † (t, 0)f (Γ, 0)

(7)

75

The evolution of ρ(r, t) can then be expressed either in terms of the evolution of the proba-

76

bility density f (Γ, t) or, via the steps below, in terms of the implicit time dependence of the

77

microscopic density field: Z

ρ(r, t) ≡ hˆ ̺(r|Γ)it = dΓˆ ̺(r|Γ)f (Γ, t) Z i h = dΓˆ ̺(r|Γ) Sˆ† (t, 0)f (Γ, 0) Z i h ˆ 0)ˆ ̺(r|Γ) f (Γ, 0) = dΓ S(t, Z = dΓˆ ̺t (r)f (Γ, 0) = hˆ ̺t (r)i0 5


(8)


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where ̺ˆt (r) ≡ ̺ˆ(r|Γt (Γ)) and the subscripts t and 0 of the angular brackets indicate aver-

79

age with respect to f (Γ, t) and f (Γ, 0), respectively. The result above is nothing but the

80

celebrated Onsager-Kubo formula 1,2,11,12 .

81

Let us now focus on ρ(r, t0 ), the local density field at some initial time t0 = 0. To connect

82

with the probabilistic framework of the previous section, let us observe that the microscopic

83

density ̺ˆ is a stochastic field. Now, let us consider an ensemble of systems described in

84

the phase space Γ by the canonical probability density. Let us also introduce a Maxwell

85

daemon imposing at time t0 a specific density profile ̺(r) as a realization of the random field

86

̺ˆ. The constrained equilibrium joint probability density for the set of stochastic variables

87

W = {ˆ ̺, Γ}, is fW [̺, Γ] =

88

89

90

1 e−βHN (Γ) Y δ [ˆ ̺t0 (r) − ̺(r)] h3N N ! ZN r∈V

(9)

where the product is meant in the functional sense, i.e. over the continuum set {Γ, ̺ˆ(r), ∀r ∈ V }, where V is the volume enclosing the system. The normalization of the joint distribution fW [̺, Γ] in Eq. (9) is Z

D̺

Z

dΓ

1 e−βHN (Γ) Y δ [ˆ ̺t0 (r) − ̺(r)] = 1 h3N N ! ZN r∈V

(10)

91

where the first integral is a functional integral over all possible density profiles. The marginal

92

probability density of the stochastic variable ̺ˆ is

f̺ˆ[̺] =

93

Z

1 e−βHN (Γ) Y δ [ˆ ̺t0 (r) − ̺(r)] dΓ 3N h N ! ZN r∈V

(11)

a functional in ̺ (i.e. a function of all values of ̺ for r ∈ V ), and, of course, Z

D̺

1 e−βHN (Γ) 1 e−βHN (Γ) Y . δ [ˆ ̺ (r) − ̺(r)] = t 0 h3N N ! ZN h3N N ! ZN r∈V

6


(12)

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The conditional probability density of the stochastic variable Γ given ̺(r), a particular

95

realization of the field ̺ˆ, is then Q e−βHN (Γ) r∈V δ [ˆ ̺t0 (r) − ̺(r)] fW [̺, Γ] Q fc [ Γ|̺ ] = =R , f̺ˆ[̺] ̺t0 (r) − ̺(r)] dΓe−βHN (Γ) r∈V δ [ˆ

96

97

(13)

ˆ t (r) ≡ a functional on ̺. With these definitions the average of any microscopic observable O PN ˆ O(r|Γ t (Γ)) = i=1 O(Γt )δ(r − Ri (t)) conditioned on ̺(r) is ˆ t (r)|̺iΓ = hO

Z

ˆ t (r) fc [ Γ|̺ ] . dΓO

(14)

98

ˆ t=t0 (r) = ̺ˆt0 (r) the equation above reduces simply to hˆ In the specific case O ̺t0 (r)|̺iΓ = ̺(r).

99

An intuitive way to understand the meaning of the functionals introduced above is the

100

following. Let us discretize the volume V over a grid of L points. For a finite grid we

101

102

can replace the function ̺(r) with the L−dimensional vector ̺ ≡ (̺1 , ̺2 , . . . , ̺L ) and the continuum products in the above equations with Y

r∈V

δ [ˆ ̺(r|Γt (Γ)) − ̺(r)] ≈

L Y ℓ=1

δ [ˆ ̺ℓ (Γt ) − ̺ℓ ]

(15)

103

where ̺ˆℓ (Γt ) is defined as the average of the microscopic field over the volume vℓ of the

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tessellating cell around the ℓ-th grid point 1 ̺ˆℓ (Γt ) = vℓ

105

106

107

Z

d3 rˆ ̺(r|Γt (Γ))

(16)

vℓ

R With this discretization, the functional integral D̺ becomes the L-dimensional integral R d̺1 d̺2 . . . d̺L . The functional interpretation is then recovered in the limit in which the

grid spacing goes to zero together with L → ∞.

7



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108

The statistical mechanics versus the macroscopic approach

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ˆ t′ (r′ ) ≡ O(r ˆ ′ |Γt′ (Γ)) at time t′ and position Let us consider again a generic microscopic field O

110

r′ , parametric in Γ. The conditional average of this observable at given ̺(r) is (see Eq.(14) ) ˆ t′ (r )|̺iΓ = hO ′

Z

ˆ t′ (r′ )fc [Γ|̺ ] . dΓO

(17)

111

We can now make a fundamental observation. When the continuum approach is valid,

112

Eq.(17) is in fact the microscopic definition of the evolution of the classical field O(r′ , t′ |̺) in

113

a continuum mechanics (or hydrodynamic) description. This evolution is now expressed as

114

ˆ t′ (r′ ) over the statistical the time-dependent average of the corresponding microscopic field O

115

ensemble conditioned on the initial macroscopic state ̺(r). Thus: ˆ t′ (r′ |̺)iΓ . O (r′ , t′ |̺) = hO

116

(18)

This is the main result of this paper and it has two key consequences.

117

The first one stems from the following observation. Let us multiply the conditional

118

average given in Eq.(17) by the imposed density profile ̺(r). Averaging over the probability of

119

the density profile itself, we obtain, by definition, the correlation function of the macroscopic

120

fields ′

′

h̺(r)O (r , t |̺)i̺ˆ ≡ 121

Z

D ̺f̺ˆ[̺] ̺(r) O (r′ , t′ |̺) .

(19)

Following the steps in Eq. (5) we also obtain ˆ t′ (r′ |̺)iΓ i̺ˆ = hˆ ˆ ′ |Γt′ (Γ))iW h̺(r)O (r′ , t′ |̺)i̺ˆ = h ̺(r)hO ̺(r|Γ)O(r

(20)

122

ˆ which gives the relation between the correlation function of the microscopic observable O

123

with the microscopic field ̺ˆ over the statistical ensemble and the correlation function of the 8


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124

corresponding, macroscopic, local fields over the (marginal) distribution of the fluctuating

125

field. This is the result we were looking for in the present work.

126

The second important consequence of Eq.(18) is that it originates a rigorous and prac-

127

tical algorithm for atomistic simulations. We postpone illustrating this point until the next

128

section.

129

ˆ t′ (r′ ) is the microscopic density field at t > t0 , Let us consider now the case in which O

130

where t0 is the initial time at which the profile ̺(r) = ρ(r, t0 ) is imposed. In terms of the

131

132

Liouville operator L ˆ t′ (r′ ) ≡ ̺ˆt′ (r′ ) = eiL(t′ −t0 ) ̺ˆt0 (r|Γ) . O

(21)

hˆ ̺t′ (r′ )ˆ ̺t0 (r)iW = h hˆ ̺t′ (r′ |̺)iΓ ̺(r)i̺ˆ = h ρ(r′ , t′ |̺)ρ(r, t0 )i̺ˆ

(22)

From Eq.(20) we obtain

133

Note that Eq.(18) and Eq.(22) are not based on the assumptions necessary for the validity of

134

the hydrodynamical calculations (local thermodynamic equilibrium, etc). The only require-

135

ment for these relationships is that the evolution of the macroscopic density be exact. In

136

the hydrodynamic limit it is assumed that the hydrodynamic evolution represents the exact

137

dynamics of the macroscopic fields so, in this limit, the right and left hand sides of Eq.(22)

138

can be identified.

139

As a side remark, note also that our results are valid irrespective of the size of the grid

140

to be used in the practical implementation of the continuum products of Eq. (15). In this

141

sense they are much more general than the equivalence between correlation functions of

142

microscopic observables and correlation functions of hydrodynamic fields, since we have not

143

invoked the hydrodynamic limit in our derivation. In particular, if the tessellating cells of

144

the discrete grid are large enough and the time span of the correlation long enough, our

145

averaging procedure provides the (k → 0, ω → 0) limit of hydrodynamics. In this situation 9



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146

the marginal probability f̺ˆ[̺] becomes the probability of the thermodynamic fluctuations,

147

i.e. we have found back the proper meaning of Einstein fluctuation theory.

148

Comments and conclusions

149

The result demonstrated in this paper opens the way for simulating by Molecular Dynamics,

150

rigorously, the dynamical behavior of fluids or, more generally, continuous systems. It also

151

provides a practical algorithm that can be illustrated as follows. According to the results of

152

the previous section, the macroscopic space- and time- dependent behaviors can be computed

153

as the average over independent trajectories of the microscopic system, evolved in time by the

154

usual Hamiltonian. The initial conditions for these trajectories must, however, be sampled

155

from the conditional probability density in Eq. (13) that represents the macroscopic state

156

of the system at the initial time. Setting conditions on macroscopic fields in a molecular

157

dynamics simulation requires some special care. When we look at hydrodynamical processes,

158

the ensemble to be realized is described by a conditional probability density imposing a

159

macroscopic condition expressed as a field-like observable. This sampling must then be

160

implemented using appropriate methods such as, e.g., the restrained Molecular Dynamics

161

approach 13 or, equivalently, the Blue Moon approach. 14,15 Once this is done we have a sample

162

of microscopic states that satisfy the requested macroscopic conditions. By dynamically

163

evolving these states and averaging at time t, we can get the dynamical evolution of the

164

associated macroscopic field. This is precisely the realization in simulation of the idea of

165

Onsager 1 as explained by Kubo, 2,3 which we have discussed elsewhere. 16–20

166

To provide a specific example, let us briefly summarize the, simpler, algorithm based on

167

the Restrained Molecular Dynamics approach for the case in which a restriction is imposed

168

on a field, e.g., on the microscopic density ̺ˆt0 (r|Γ). For a chosen shape of the density

169

170

profile ̺(r) we have to impose the condition ̺ˆt0 (r|Γ) = ̺(r) on all accessible points r ∈ V . Numerically we cannot deal directly with the continuous variable r and we need to discretize

10


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171

the volume available over a finite mesh of points. For example, adopting the discretization

172

in Eq. (16), we can subdivide the volume in elementary cubic cells, so that the reference

173

point rℓ coincides with the center of the ℓ-th cell, ℓ = 1, . . . , L. The discretized microscopic

174

field at each point rℓ is then defined as the average ξ of the microscopic density ̺ˆ(r|Γ) over

175

the volume vℓ of the ℓ − th cell: 1 ξ(rℓ |Γ) = vℓ

176

Z

d3 r vℓ

"

N X j=1

δ(r − Rj )

#

,

(23)

In the Restrained Molecular Dynamics approach the condition ξ(rℓ |Γ) = ̺(rℓ ), ℓ =

177

1, 2, . . . , L is imposed by means of an additional potential term added to the Hamiltonian

178

HN (Γ) of the unrestrained system. The restrained Hamiltonian L

kX [ξ(rℓ |Γ) − ̺(rℓ )]2 Hk (Γ) = HN (Γ) + 2 ℓ=1

(24)

179

depends on the tunable parameter k that defines the strength of the (pseudo-harmonic) re-

180

straining potential, i.e., the last term on the right-hand side of Eq. (24). This Hamiltonian

181

can be used to drive a Molecular Dynamics (or Monte Carlo) simulation at a fixed temper-

182

ature, T , generating trajectories, stochastic or deterministic, which sample the phase space

183

of the system according to the (restrained) canonical probability density at time t0 , 1 e−βHk (Γ) h3N N ! Z (k) Q βk 2 1 e−βHN (Γ) · ℓ e− 2 [ξ(rℓ |Γ)−̺(rℓ )] f (k) (Γ, ̺) = 3N ≡ h N! ZN · f (k) (̺) f (k) (̺)

fc(k) (Γ|̺) =

184

185

(25)

where the condition on ̺ˆ is imposed in terms of the values at each mesh point. In the R equation above, Z (k) = dΓe−βHk (Γ) h3N N ! and ZN are the canonical partition functions

11



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and f (k) (̺(r1 ), ̺(r2 ), . . .) =

Z

dΓ

(

e−βHN (Γ) Y − βk [ξ(rℓ |Γ)−̺(rℓ )]2 e 2 h3N N ! ZN ℓ

)

(26)

187

For a given value k the restrained algorithm samples the k-dependent conditional probability

188

density of Γ of Eq.(25). Using the definition o n a a lim √ exp − (y − y˜)2 = δ(y − y˜) a→∞ 2 2π

189

it can be seen that for βk → ∞ this becomes indeed the ensemble associated with the

190

macroscopic condition. In practice, in this approach 13 the bias is tuned to high enough

191

values of the parameter k to provide a good approximation of the conditional probability.

192

The main ingredients of the algorithm are sketched in Figure 1. The stationary restrained

193

Molecular Dynamics trajectory is represented as the blue curve. Independent points along

194

this trajectory (the open white circles in Figure 1) are initial configurations sampling the

195

probability density at time t0 . The hydrodynamic evolution can be followed by averaging

196

the microscopic behavior of the system over the independent trajectories (the black curves in Figure1) generated by the Hamiltonian HN (Γ). ˆ t (r) O

̺ˆ

( r| t0

ˆ t (r) O

Γ) ˆ t (r) O

ˆ t (r) O

ˆ t (r) O

r ̺ˆt 0(

|Γ)

ˆ t (r) O

ˆ t (r) O

Figure 1: Graphical representation of the algorithm: the black curves represent the ensemble of unrestrained trajectories starting from independent points, the open white circles, sampled along the stationary Restrained Molecular Dynamics trajectory, the light blue curve. 197

198

The approach described above has already been used in a number of applications. In 12


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199

particular, we recall the study 18 of the relaxation of a curved non equilibrium profile of the

200

macroscopic interface between two immiscible fluids and the study 19 of the Joule-Thomson-

201

like expansion of a fluid into a vacuum region. In the latter, the dynamical relaxation of the

202

local density profiles, together with the behaviors of the momentum current and the local

203

temperature, are compared with the field values obtained from the corresponding solution of

204

the hydrodynamics equations. Their agreement demonstrates the potential of the approach.

205

To conclude, we have shown the profound connection that links the evolution of the

206

macroscopic quantities in the hydrodynamic, or continuum mechanics, description with suit-

207

able averages of the corresponding microscopic observable. These averages are obtained by

208

propagating initial conditions sampled from the constrained conditional probability density

209

associated to the macroscopic state at the initial time. The same link exists between the

210

correlations of hydrodynamics quantities and those of the corresponding microscopic observ-

211

ables. The understanding that the hydrodynamic behavior can be described as the time

212

evolution, conditioned on a specific macroscopic (initial) condition, enables to follow the re-

213

laxation process, 18,19 or more generally the hydrodynamic evolution, 17 via the time evolution

214

of microscopic averages over the conditional probability density of the initial ensemble. The

215

technical problems involved with the critical step in the procedure, i.e the ability to repre-

216

sent the initial macroscopic condition in terms of an ensemble of microscopic states to be

217

sampled numerically, can be tackled successfully by, e.g., the restrained Molecular Dynamics

218

(or Monte Carlo) approach. This framework then makes it possible to study complex hydro-

219

dynamic phenomena using only the fundamental laws of Statistical Mechanics, i.e. without

220

recourse to ‘local smoothing’ or other empirical recipes typical of continuum hydrodynamic

221

theories. We are at present developing the implications of this approach.

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Acknowledgement

223

It is a pleasure to dedicate this paper to Bruce C. Garrett for his beautiful work which

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has contributed so much to our field. The authors acknowledge the italian MIUR PRIN-

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2012NNRKAF_004 Grant for financial support.

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References

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Graphical TOC Entry ˆ t (r) O

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(r ̺ˆt 0

r ̺ˆt 0(

ˆ t (r) O

|Γ) ˆ t (r) O

ˆ t (r) O

ˆ t (r) O

|Γ)

ˆ t (r) O

ˆ t (r) O The restrained trajectory with the ensemble of independent ones

17


Probabilistic Derivation of Spatiotemporal Correlation Functions in the

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