Dynamic principle for ensemble control tools A. Samoletov1, 2, ∗ and B. Vasiev1, †
arXiv:1702.08399v1 [physics.data-an] 27 Feb 2017
1
University of Liverpool, Department of Mathematical Sciences, Peach Street, Liverpool, L69 7ZL, UK 2
Institute for Physics and Technology, Donetsk, Ukraine (Dated: February 28, 2017)
Abstract Dynamical equations describing physical systems at statistical equilibrium are commonly extended by mathematical tools called “thermostats”. These tools are designed for sampling ensembles of statistical mechanics. We propose a dynamic principle for derivation of stochastic and deterministic thermostats. It is based on fundamental physical assumptions such that the canonical measure is invariant for the thermostat dynamics. This is a clear advantage over a range of recently proposed and widely discussed in the literature mathematical thermostat schemes. Following justification of the proposed principle we show its generality and usefulness for modelling a wide range of natural systems. PACS numbers: 02.70.Ns, 05.40.−a, 05.45.−a, 05.70.Ln, 45.50.−j
∗
[email protected];
[email protected]
†
[email protected]
1
Analysis of molecular systems is an essential part of research in a range of disciplines in natural sciences and in engineering [1–3]. As molecular systems affected by environmental thermodynamic conditions, they are studied in the context of statistical physics ensembles. Methods of dynamical sampling of the corresponding probability measures are important for applications and they are under extensive study and development [4–14]. The traditional application of thermostats is molecular dynamics (MD), that is sampling of equilibrium systems with known potential energy functions, V (q), where q is a system configuration. However, being able to sample equilibrium ensembles would imply that we are also able to sample arbitrary probability measures. Indeed, as an alternative to the common MD practice, one may use a probability density σ(q) to define the potential function as V (q) = −β −1 ln σ(q). Thermostats embedded in dynamical equations bring in the so-obtained dynamics a rich mathematical content. Such dynamical systems with an invariant probability measure have become increasingly popular for mathematical studies in a wide-range of applications including investigation of non-equilibrium phenomena [6, 15–21], mathematical biology models [22–25], multiscale models [26–31], Bayesian statistics and Bayesian machine learning applications [29, 32–34], superstatistics [35, 36]. Here, we present a unified approach for derivation of thermostats sampling the canonical ensemble. The proposed principle is based on clear physical arguments that facilitate understanding of thermostat schemes, and elucidate physics of the Nos´e-Hoover (NH) and the Nos´e-Hoover-Langevin (NHL) dynamics in particular. Our main result is that the proposed dynamic principle allows to build a plethora of thermostats, stochastic as well as deterministic, including those previously proposed. The presented method can, in principle, be used to develop dynamic temperature control tools such that differ from the classical NH type dynamics, and can be extended to non-physical probability measures. We expect that it can also be adjusted to arbitrary probability measures. Classical mechanics and equilibrium statistical physics are adequately described in terms of the Hamiltonian dynamics. Theoretical thermostat schemes involve modified equations of motion where different temperature control tools are included. The modified equations can be deterministic as well as stochastic [1–14, 23, 37–39]. Some of recently proposed thermostats [4, 9–11] combine a modified deterministic dynamics with stochastic perturbations. This combination ensures ergodicity and demonstrate a “gentle” perturbation of the physical 2
dynamics that is often desired [4, 10]. Consider a dynamical system S consisting of N particles in d-dimensional space (N = dN degrees of freedom) described by the Hamiltonian function H(x), where x = (p, q) is a N point in the phase space M = R2dN , p = pi ∈ Rd i=1 are momentum variables and q = N qi ∈ Rd i=1 are position variables. The Hamiltonian dynamics has the form, x˙ = J ∇H(x) in the phase space M, where J is the symplectic unit. The canonical ensemble describes a system in contact with the heat bath Σ (which is an energy reservoir permanently staying at the thermal equilibrium). This system may exchange energy with the heat bath only in the form of heat. In the canonical ensemble the temperature T is fixed but the system energy E is allowed to fluctuate. The canonical distribution has the form ρ∞ (x) ∝ exp [−βH (x)], where β = (kB T )−1 , kB is the Boltzmann constant. In average along an ergodic trajectory hE(t)i = E(T ) = const. Rate of energy exchange between system S and thermal bath Σ depends on the temperature T . Note, that Hamiltonian system is unable to sample the canonical distribution since there is no energy exchange between the system and the heat bath. Thus, to describe the heat transfer we need to modify Hamiltonian equations of motion coming to a non-Hamiltonian dynamics [40]. Suppose x˙ = G(x) is a law of motion ˙ and H(x) = G(x) · ∇H(x) is the rate of energy change such that hG(x) · ∇H(x)i = 0, that is the energy is constant at the average. Denote G(x) · ∇H(x) as Z(x), hZ(x)i = 0. Definition of Z(x) allows a certain freedom. To further elaborate this reasoning, we need some preliminaries. We say that F (x, β), hF (x, β)i = 0 for all β > 0, is the microscopic temperature expression (TE) and write F (x, β) ∼ 0 here and thereafter. As an example, Fk (p, β) = 2K(p)β − N is kinetic TE and Fc (q, β) = (∇V (q))2 β − ∆V (q) is configurational TE (e.g. [41]), where K(p) + V (q) = H(x). Various TEs, in the canonical ensemble, can be obtained in the following manner. Suppose P n F (x, β) is polynomial in β, F (x, β) = 2L+1 n=0 ϕn (x)β ∼ 0, where L ∈ Z≥0 and functions
x ∈ M, are subject to specification. Rewrite F (x, β) in the form F (x, β) = {ϕn (x)}2L+1 n=0 , PL 2k ∼ 0 for all β > 0. Thus, {(ϕ2k (x) + βϕ2k+1(x))} ∼ 0 for all k=0 (ϕ2k (x) + βϕ2k+1 (x)) β
k ∈ {0, 1, . . . , L}, and suffice it to specify the basic expression, F (x, β) = β ϕ1 (x)+ϕ0 (x) ∼ 0. Substituting ϕ(x)∂i H(x) for ϕ1 (x), where ϕ(x) is an arbitrary function, and utilizing the identity, ∂i e−βH(x) = −β∂i H(x)e−βH(x) for all i = 1, . . . , 2dN and x ∈ M, where ∂i ≡ ∂/∂xi , we get the relationship, ∂i ϕ(x)+ϕ0 (x) ∼ 0. Then, excluding ϕ0 (x) from the basic expression, 3
we arrive at the set of basic TEs, β ϕ(x)∂i H(x) − ∂i ϕ(x) ∼ 0 for each and every xi in M. This set can be represented in compact form. Suppose ϕ(x) is an arbitrary vector field on M. Therefore, we have F(x, T ) = ϕ(x) · ∇H(x) − kB T ∇ · ϕ(x) ∼ 0,
(1)
where kB T = β −1 . The similar TE was considered in [42]. In what follows we focus our attention on this particular expression. However, more general TEs are allowed, e.g. vector fields F (x, β) = β ∇H(x) × ϕ(x) − ∇ × ϕ(x) ∼ 0, and so on. For the sake of completeness P we note that F (x, β) = Lk=0 β ϕ(k) (x) · ∇H(x) − ∇ · ϕ(k) (x) β 2k ∼ 0 for all L ∈ Z≥0 , where {ϕ(k) (x)} is a set of vector fields such that |ϕ(x)| exp[−βH(x)] → 0 as |x| → ∞.
Although expression (1) involves arbitrary vector field resulting in existence of infinite number of TEs, they all are equivalent from the thermodynamic perspective. However, the time interval required to achieve a prescribed accuracy in hF (x, β)i = 0 may differ for particular, e.g. kinetic and configurational, expressions [43]. In general, physical systems are often distinguished by multimodal distributions and by existence of long-lived metastable states. Thus, their dynamics is characterized by processes going on a number of timescales. We presume that different TEs can be associated with dynamical processes running on various time scales and then applied to a multiscale models. Now we introduce the following principle: Let F (x, β) ∼ 0 be a TE; then, in the canonical ensemble, ∇H(x) · x˙ ∝ F (x, β).
(2)
Relationship (2) states that the rates of dynamical fluctuations in energy and in TE are proportional, both are zero in average and there is no energy release along a whole trajectory in M. Factor of proportionality depends on the strength of interaction between the system S and thermal bath Σ, and on TE. Relationship (2) gives a necessary condition. Our aim is to show that this relationship allows construction of stochastic and deterministic thermostats. That is, if dynamical equations satisfy certain general physical assumptions and compatible with (2), than the canonical measure is invariant for the dynamics. To address this problem, let us consider the exchange of energy between system S and thermal bath Σ. Any system placed in the heat bath should in some extent perturb it and be affected by backward influence of this 4
perturbation. Thus, there exist subsystem Sad of Σ such that Sad is involved in joint dynamics with S. The rest of the heat bath is assumed to be unperturbed, permanently staying at thermal equilibrium. This is an approximation that is based on separation of relevant characteristic time scales. For instance, method of Brownian dynamics is based on an assumption that characteristic time scales of processes in S and Σ are considerably different and the system S does not perturb Σ at all. If the time scale is refined (which is of particular importance for small systems) then we have to take into account joint dynamics of S and Sad . We will show that this case is closely associated with the NHL [9, 10] and the classical NH dynamics [38, 39]. Thus, we have two cases: (A) the system S doesn’t perturb the thermal bath. The factor of proportionality in (2), λ, is a constant. In this case, there is no new dynamic variable (or variables) and the thermal bath can be taken into account only implicitly via stochastic perturbations and constant parameters (similar to the Langevin dynamics); (B) S system perturbs a part of the thermal bath Σ (Sad ) while Sad is in a thermal contact with the rest of Σ. We assume that at the thermal equilibrium systems S and Sad are statistically independent, and there is no direct energy exchange between S and Σ. Now we consider cases A and B in detail. A. Suppose
∇H(x) · x˙ = λF(x, T ), where λ is a constant. Without loss of generality,
we can consider modified Hamiltonian dynamics in the form, x˙ = J ∇H(x) + ψ(x, λ), and consequently: ∇H(x) · ψ(x, λ) = λF(x, T ),
(3)
where the vector field ψ(x, λ) is to be found. Since the thermal bath does not appear in this equation explicitly, only stochastic thermal noise may be involved in the dynamics. To find ψ, we introduce 2N -vector of independent thermal white noises, ξ(t), that is hξ(t)i = 0, hξi (t)ξj (t′ )i = 2λkB T δij δ(t − t′ ), and a vector field on M, Φ(x), such that hξ(t) · Φ(x)i = λkB T h∇ · ϕ(x)i, where h· · · i is the Gaussian average over all realizations of ξ(t). Using E P D δxi (t) δxi (t) Novikov’s formula [44], we get hξ(t) · Φ(x)i = (∂ Φ ) i k δξk (t) λkB T . Suppose δξk (t) = i,k ζi (x)δik , where the vector field ζ(x) is such that each component ζi (x) does not depend
on xi , that is ∇ ◦ ζ(x) = 0; ◦ denotes the component-wise (Hadamard) product of two vectors and 0 is the null vector. Then ∇ · ϕ(x) = ∇ · (ζ(x) ◦ Φ(x)). Thus, we get ϕ(x) = ζ(x) ◦ Φ(x) and it follows that Φ(x) = ζ −1 (x) ◦ ϕ(x), where ζ −1 (x) is the vector field such 5
that ζ −1 (x) ◦ ζ(x) = 1. Assuming ϕ(x) = η(x) ◦ ∇H(x), where η(x) ≡ ζ(x) ◦ ζ(x), we get ψ(x, λ) = −λη(x) ◦ ∇H(x) + ζ(x) ◦ ξ(t) Therefore, the modified Hamiltonian dynamics takes the form of stochastic differential equation (SDE), x˙ = J ∇H(x) − λη(x) ◦ ∇H(x) + ζ(x) ◦ ξ(t).
(4)
Note that symbol ◦ does not refer to the Stratonovich calculus. Equation (4) is SDE with additive noise and the corresponding dynamics is Markovian. An example of this form of stochastic equations of motion was considered in [11]. The Fokker-Planck equation (FPE) corresponding to SDE (4) has the form ∂t ρ = F ∗ ρ, where F ∗ ρ = −J ∇H(x) · ∇ρ + λ∇ · [η(x) ◦ ∇H(x) ρ] +λkB T ∇· [η(x) ◦ ∇ρ]. Invariant probability density for dynamics (4) is determined by the equation F ∗ ρ = 0. It is expected that this is a unique invariant density [10, 45]. We claim that for the defined above vector field ζ(y) the canonical density, ρ∞ ∝ exp [−βH (x)], is invariant for stochastic dynamics given by (4), that is F ∗ ρ∞ ≡ 0. The proof is by direct calculation. The Langevin equation is a particular case of (4). For example, let x = (p, q) ∈ M = R2 and H(x) = p2/2m + V (q). Then if ζ = (1, 0), then p˙ = −V ′ (q) − λp/m + ξ(t), q˙ = p/m; if ζ = (0, 1), then p˙ = −V ′ (q), q˙ = p/m − λV ′ (q) + ξ(t). B. Let Sad be associated with an even-dimensional phase space Mad , the Hamiltonian function h(y), y ∈ Mad , and the Hamiltonian dynamics, y˙ = J y ∇y h(y), where J y is the symplectic unit. We can assume without loss of generality that modified Hamiltonian dynamics of systems S and Sad has the form, x˙ = J x ∇x H(x) + ψ(x, y), y˙ = J y ∇y h(y) + ψ ∗ (y, x), where ψ(x, y) and ψ ∗ (y, x) are vector fields on M and Mad correspondingly, and the followE
T
ing chain of energy exchange: S ! Sad ! Σ. To derive the deterministic NH dynamics, 6
we ignore a heat exchange between Sad and the thermal bath Σ (that is the relationship ∇y h(y) · ψ ∗ = λF(y, T ) leading to the stochastic dynamics as discussed above) and relate a temperature control tool (TE for system S) to dynamics of Sad , ∇y h(y) · ψ ∗ (y, x) = −G(y)F(x, T ), where G(y) is a function of y ∈ Mad . This relationship is valid for arbitrary h(y). Assume that G(y) = Q(y) · ∇h(y), where Q(y) is a vector field on Mad . Then we have ψ ∗ (y, x) = F(x, T ) Q(y). If the total system S + Sad is isolated, then its energy is constant, dH(x) = −dh(y), and we arrive at the relationship: ∇x H(x) · ψ(x, y) = G(y)F(x, T = 0) = (Q(y) · ∇y h(y)) (ϕ(x) · ∇x H(x)) . This relationship is valid for arbitrary H(x). Thus, we have ψ(x, y) = (Q(y) · ∇h(y)) ϕ(x). Consequently, x˙ = Jx ∇x H(x) + (Q(y) · ∇y h(y)) ϕ(x),
(5)
y˙ = J y ∇y h(y) − F(x, T ) Q(y). These equations are useful in a dynamical sampling context only if the systems S and Sad are statistically independent at the thermal equilibrium, so that ∇x H(x)· x˙ ∼ 0 and ∇y h(y)· y˙ ∼ 0 hold together. Direct calculations show that these conditions are simultaneously satisfied if and only if Q(y) is an incompressible vector field (i.e. ∇y · Q(y) = 0 for all y ∈ Mad ) and |Q(y)| exp[−βh(y)] → 0 as |y| → ∞. The system (5) represents NH dynamics. The Liouville equation corresponding to the dynamical system (5) has the form ∂t ρ = −L∗ ρ, where L∗ ρ = ∇x · (xρ) ˙ + ∇y · (yρ). ˙ Invariant probability densities for (5) are determined by the equation L∗ ρ = 0. We claim that if Q(y) and ϕ(x) are the defined above vector fields, then the canonical density ρ∞ ∝ exp [−βH (x)] · exp [−βh (y)] is invariant for the NH dynamics (5), that is L∗ ρ∞ = 0. The proof is by direct calculation. Now we take into account the influence of the heat bath Σ on Sad dynamics given by the relation ∇y h(y) · ψ∗ = λF(y, T ). Following the arguments used to derive SDE (4), we arrive 7
at the stochastic dynamics: x˙ = J x ∇x H(x) + (Q(y) · ∇y h(y)) ϕ(x), y˙ = J y ∇y h(y) − F(x, T ) Q(y) − λη(y) ◦ ∇y h(y) +ζ(y) ◦ ξ(t).
(6)
This is the NHL dynamics [9, 10]. Here Q(y), ϕ(x), η(y), and ζ(y) are the previously defined vector fields. FPE corresponding to (6) has the form ∂t ρ = F ∗ ρ, where F ∗ ρ = −J x ∇x H(x) · ∇x ρ − J y ∇y h(y) · ∇y ρ − [Q(y) · ∇y h(y)] ∇x · [ϕ(x)ρ] + F(x, T )∇y · [Q(y)ρ] +λkB T ∇y · [η(y) ◦ ∇y ρ] + λ∇y · [η(y) ◦ ∇y h(y)ρ]. Invariant probability density for the SDE (6) is determined by the equation F ∗ ρ = 0. We claim that if Q(y), ϕ(x), and ζ(y) are the vector fields defined above, then the canonical density, ρ∞ ∝ exp [−βH (x)] · exp [−βh (y)], is invariant for NHL dynamics (5), that is F ∗ ρ∞ = 0. The proof is by direct calculation. We expect that this dynamics is ergodic [10, 45]. Classical NH [38, 39] and NHL [9–11] dynamical equations are particular examples of equations (5) and (6) correspondingly. For example, by substituting
ζ 2/2Q
for h(y), y =
(ζ, η) ∈ R2 , (−Q, 0) for Q(y) and (p, 0) for ϕ(x) in (5) we get classical NH equations [38, 39]. All dynamic thermostats found in the literature are described by the dynamical equations (4), (5), and (6), even though we utilized a special case of TE. In conclusion, we note that the dynamic principle proposed in this work is based on fundamental physical assumptions (the key requirement is statistical independence of systems S and Sad at thermal equilibrium) and offers a general way to develop stochastic and deterministic statistical ensemble control tools. The method was illustrated with the special TE which allowed to derive a wide range of stochastic and deterministic dynamical systems with invariant canonical measure. We also note that if the physical assumptions which we used to derive our scheme are satisfied, then the canonical measure is automatically invariant for the dynamics. The presented scheme is flexible and it is expected to be useful in development of dynamic control tools for other ensembles including those that have non-physical statistics. 8
ACKNOWLEDGMENTS
This work has been supported by the BBSRC grant BB/K002430/1 to BV.
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