Generating generalized distributions from dynamical simulation
Eric J. BarthDepartment of Mathematics, Kalamazoo College, Kalamazoo, Michigan 49006 Brian B. Laird
Department of Chemistry, University of Kansas, Lawrence, Kansas 66045 Benedict J. Leimkuhler
Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, United Kingdom
共Received 3 May 2002; accepted 10 January 2003兲
We present a general molecular-dynamics simulation scheme, based on the Nose´ thermostat, for sampling from arbitrary phase space distributions. We formulate numerical methods based on both Nose´–Hoover and Nose´–Poincare´ thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that is designed to accelerate sampling in molecular systems. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1557413兴
I. INTRODUCTION
Molecular-dynamics 共MD兲 computer simulation is a widely used tool in biology, chemistry, physics, and materi-als science.1,2Much of the power in the technique lies in the ability to generate phase-space trajectories weighted accord-ing to a relevant statistical-mechanical distribution. In the first MD simulations, straightforward integration of the equa-tions of motion for the system under study yielded energy conserving trajectories that, assuming ergodicity, generated microcanonical 共constant NVE兲 equilibrium distributions of phase-space configurations. Later, to better mimic experi-mental conditions, a variety of MD techniques were devel-oped that generate other standard statistical-mechanical dis-tributions, such as canonical 共NVT兲,3–5 isothermal–isobaric (N PT),6,7 and grand-canonical 共VT兲.8 Recently, however, there has been growing interest in the simulation of systems with distributions that go beyond textbook statistical me-chanical ensembles. For example, a variety of molecular-dynamics methods focusing on the simulation of systems obeying Tsallis statistics9 have been developed by Plastino and Anteneodo,10Andricioaei and Straub,11and Fukuda and Nakamura.12 In this work we outline a general molecular-dynamics scheme, based on the Nose´ thermostat,3,13to gen-erate configurations according to an arbitrary phase-space distribution.
A primary motivation for the development of algorithms for the generation of nonstandard distributions is the need for methods that accelerate the configurational sampling of systems. Many systems are not sufficiently ergodic on the time scale of standard molecular-dynamics simulations to en-sure the convergence of statistical averages. This is espe-cially true of macromolecules, biomolecules, and amorphous materials. Over the past decade, a number of methods have been developed to enhance sampling in MD. Berne and Straub14 have recently written an excellent review of new sampling methods. Central to these approaches has been the recognition that high activation barriers cause a bottleneck
in phase space, rendering transitions between states unlikely. A common thread among many methods is the systematic deformation of the potential 共or total兲 energy surface to ac-celerate barrier crossing, either by lowering the barriers or raising the potential valleys. From a statistical mechanical perspective, such energy modifications induce a correspond-ing modification in the phase-space distribution by en-hancing the statistical weight of configurations in the vicinity of energy barriers. Explicit knowledge of the modified sampling distribution allows for statistical reweighting of the computed trajectories to achieve averages in the original ensemble.
The simplest method for enhancing sampling is to scale the full Hamiltonian by some factor less than unity. This is equivalent to performing the simulation at a higher tempera-ture. If averages are desired at temperature T, isothermal MD simulations can be carried out at some higher temperature T*, with averages at the original temperature computed by reweighting the probability of each configuration by a factor of exp关((1/kT)⫺(1/kT*))H( p,q)兴. A significant disadvan-tage with such temperature boost approaches is that, unless the boost is sufficiently small, low energy configurations are not visited with a frequency large enough to obtain accept-able statistics. A related approach, multicanonical MD, is based on a Monte Carlo technique of the same name15 and uses preliminary high temperature trajectories to construct a distribution that is nearly flat in the position coordinates, allowing nearly uniform sampling of coordinate space in subsequent simulations. Multicanonical MD has been dem-onstrated to accelerate conformational sampling in model polypeptides16and atomic clusters.
Another approach is Voter’s hyperdynamics,17,18 which employs a ‘‘boost potential’’ to reduce the sampling prob-ability in low energy regions, thereby accelerating barrier crossing due to diminished relative energetic cost. With boost potentials chosen to leave the potential energy in bar-rier regions unchanged, transition state theory arguments can 5759
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be used to obtain good transition rate statistics. However, the low-energy sampling problem remains as in high tempera-ture dynamics. Hyperdynamics has been used successfully in solid state systems, but the method is not generally appli-cable to liquids, where the presence of many saddle points hampers the identification of well-defined barrier regions.
An approach to enhanced sampling can also be based on a modification共typically a smoothing兲of the potential energy function. In some cases, enhanced sampling via potential modification can be realized directly without altering any equilibrium properties by a suitable coordinate transfor-mation.19 The approach of Straub and co-workers11 is of particular relevance for the current article, with Monte Carlo and MD methods based on potential energy modifica-tions that sample coordinates from alternative densities ac-cording to a formalism motivated by the nonextensive Tsallis entropy.9The Tsallis–Straub approach is easy to implement, amounting to a simple modification of the interaction forces according to the gradient of an effective potential. Recently, a more direct application of Tsallis entropy to MD was sug-gested by Plastino and Anteneodo.10Based on the idea of an effective Hamiltonian, these authors showed that canonical sampling 共using the dynamical approach of Kusnezov et al.20兲with respect to the effective Hamiltonian is equiva-lent to Tsallis sampling. Significantly, this work considered only the Tsallis regime in which coordinate sampling is re-stricted to low energy regions. In a similar very recent work, Fukuda and Nakamura,12 generate Tsallis dynamics by adapting the effective Hamiltonian approach of Plastino and Antenenodo to a Nose´–Hoover thermostat.4 In addition, the possibility of extending the approach to arbitrary statistical distributions is discussed.
In this paper we present a dynamical framework for sampling from a general class of probability density func-tions, including but not limited to the Tsallis density. In order to introduce the idea of sampling from nonmicrocanonical distributions and to provide the necessary background for our generalized dynamics we discuss in Sec. II the extended Hamiltonian approach of Nose´3 to canonical共constant tem-perature兲sampling, as well as the Nose´–Hoover4and Nose´– Poincare´5 approaches for implementing real-time formula-tions of Nose´ dynamics. In Sec. III we introduce Generalized Distribution Dynamics共GDD兲and discuss the technique for two special classes of systems: those for which the position and momentum distributions are separable and those for which the phase space distribution is a function of the full Hamiltonian, and show how the Nose´ framework can be used to derive the equations of motion that produce trajecto-ries which sample from generalized distributions. In Sec. IV we present as an example the variable temperature distribu-tion for accelerating the sampling of systems with high bar-riers. Numerical experiments on a double well potential us-ing both the separable and full Hamiltonian GDD approaches are presented in Sec. V.
II. SAMPLING FROM A CANONICAL DISTRIBUTION: THE NOSE´ THERMOSTAT
In traditional 共NVE兲 MD simulation, the equations of motion corresponding to the system Hamiltonian, H(p,q),
are integrated to generate the trajectories. The trajectory is constrained to the constant energy surface, E⫽H(p,q) deter-mined by the initial values of coordinates and momenta. States in phase space along solutions are said to be sampled from the microcanonical, or constant energy, distribution ac-cording to the probability density NVE(q,p) that is
propor-tional to␦(H(q,p)⫺E), where␦is the Dirac delta function. Due in part to a desire to bring simulation into accord with laboratory experiments that are typically conducted at some fixed temperature, methods have been developed for generating trajectories which sample from the canonical, or constant temperature, ensemble according to the proba-bility density NVT(q,p), which is proportional to exp关⫺H(q,p)兴, where ⫽1/(kBT), T being the tempera-ture and kB the Boltzmann constant. In contrast to the micro-canonical case, micro-canonical sampling allows states at all ener-gies, though higher energy states have lower probabilities depending on the value of temperature T.
Although other methods exist, the most widely used techniques for generating canonically distributed trajectories in MD simulation are based on the extended Hamiltonian of Nose´,3,13 HNose´⫽p˜ TM⫺1˜p 2s2 ⫹V共q兲⫹ 2 2Q⫹gkBT ln s, 共1兲 where s and are conjugate thermostat variables, Q is a fictional thermostat mass which determines the strength of thermal coupling to the system, g⫽Nf⫹1 共with Nfbeing the number of degrees of freedom in the system兲, and p˜ is a
virtual momentum related to the actual momentum of the system by p˜⫽sp.3The equations of motion generated by the Nose´ Hamiltonian关Eq.共1兲兴are
dq d⫽M ⫺1˜/sp 2, 共2兲 dp˜ d⫽⫺ⵜV共q兲, 共3兲 ds d⫽ Q, 共4兲 d d⫽ p ˜TM⫺1˜p s3 ⫺gkBT/s. 共5兲
The Nose´ method regulates the temperature of the sys-tem through a dynamical time transformation given by d/dt⫽s, where is the Nose´ 共virtual兲 time and t is real time. The remarkable property of Nose´ dynamics is that mi-crocanonical sampling of the extended phase space 兵q,p˜,s,其 yields canonical sampling in the reduced phase space,兵q,p其, provided that the system is ergodic.
For practical calculations of averages such as velocity autocorrelation functions, it is convenient to work in a real-time implementation of the Nose´ thermostat. The most com-monly used real-time modification is due to Hoover.4Hoover recognized that one can generate a set of real-time equations of motion by making the following transformations to the Nose´ equations of motion:
共1兲 Change of variables: p⫽˜/s;p
共2兲 Time transformation: d/dt⫽s;
共3兲 Change of variables: ⫽ln s and⫽˙ .
The result is the following time-reversible system of equa-tions, known as the Nose´–Hoover 共NH兲equations,4
q˙⫽M⫺1p, 共6兲 p˙⫽⫺ⵜV共q兲⫺p, 共7兲 ˙⫽, 共8兲 ˙⫽1 Q关p TM⫺1p⫺gk BT兴, 共9兲
where g⫽Nf, the number of degrees of freedom in the
sys-tem. These equations of motion are non-Hamiltonian in form since the coordinate transformations were not canonical; however, a conserved energy does exist given by20
ENH⫽
pTM⫺1p
2 ⫹V共q兲⫹ Q2
2 ⫹gkBT. 共10兲
共Although the variable has been decoupled from the sys-tem, it is helpful to include it in the calculations so that E can be monitored as an indicator of trajectory stability.兲
Recently, Bond, Leimkuhler, and Laird5have developed an alternative real-time Nose´ thermostat scheme, the Nose´– Poincare´ method, which is Hamiltonian in form, allowing for the use of symplectic integration schemes共which have been shown to give superior stability in long time simulation21兲. This is accomplished by performing a time transformation, not to the Nose´ equations of motion as with Nose´–Hoover, but directly to the Hamiltonian using a Poincare´ time trans-formation, as follows:
HNP⫽s共HNose´⫺H0兲, 共11兲
where H0 is the initial value of HNose´. It can be easily
verified5that the phase space trajectory generated by HNP is identical to that generated by HNose´except for a time trans-formation d/dt⫽s. The Nose´–Poincare´ equations of mo-tion are q˙⫽s⫺1M⫺1p˜, 共12兲 s˙⫽s Q , 共13兲 pP⫽⫺sⵜV共q兲, 共14兲 ˙⫽s⫺2p˜TM⫺1p˜⫺gkT⫺⌬H, 共15兲 where ⌬H⫽˜p TM˜⫺1˜p 2s2 ⫹Vc共q兲⫹ 2 2Q⫹gkT ln s⫺H0. 共16兲 Note that, the exact solution to the Nose´–Poincare´ equations of motion generates trajectories that are identical to those generated by the Nose´–Hoover scheme, exactly solved. It is in the construction of approximate numerical methods that these two approaches differ.
Although we favor the Nose´–Poincare´ method in all cases, the Nose´–Hoover formalism is more familiar within
the simulation community. For this reason, in the current article, we present schemes based on both the Nose´–Hoover and Nose´–Poincare´ approaches.
In certain systems, for example those with few particles or strong harmonic components, the ergodicity assumption basic to the Nose´ approaches is not met. For these cases, the notion of Nose´–Hoover chains has been developed,22 in which the Hamiltonian is further extended with additional thermostat variables that are coupled to each other. It has been demonstrated that NH chains, with properly chosen thermostat masses, can induce the needed ergodicity so that NH dynamics provides a means of sampling from the ca-nonical distribution. We discuss NH chains further in Sec. V and in the Appendix.
III. GENERALIZED DISTRIBUTION DYNAMICS
In this section we present a dynamical scheme for sam-pling points in phase space from a general distribution func-tion F(p,q) which satisfies the properties of a probability density function in the phase space variables 兵p,q其,
冕
p,qF共p,q兲⫽1 and F共p,q兲⭓0.
In analogy to the procedure used by Plastino and Anteneodo10 to develop a MD method to generate the ca-nonical Tsallis distribution, we relate the general density to the canonical density by way of an effective Hamiltonian Heff as F共p,q兲⫽e⫺Heff, which yields Heff⫽⫺ 1 ln F共p,q兲. 共17兲
It is clear that canonical sampling with respect to the effec-tive Hamiltonian is equivalent to sampling according to the generalized probability density F. To achieve canonical sam-pling with Heffwe write the Nose´ Hamiltonian for General-ized Distribution Dynamics,
HNose´F ⫽⫺1
ln F共˜/s,qp 兲⫹ 2
2Q⫹gkBT ln s. 共18兲 From the equations of motion generated from this Nose´ Hamiltonian and after applying the transformations de-scribed in the previous section, we obtain the Nose´–Hoover GDD equations of motion, q˙⫽⫺ kBT F共p,q兲 ⵜpF共p,q兲, 共19兲 p˙⫽ kBT F共p,q兲 ⵜqF共p,q兲⫺p, 共20兲 ˙⫽, 共21兲 ˙⫽1 Q
冋
⫺kBT F共p,q兲p Tⵜ pF共p,q兲⫺gkBT册
. 共22兲Similarly, the Nose´–Poincare´ equations of motion for the GDD are
q˙⫽⫺ kT F共p˜/s,q兲 ⵜp˜/sF共p˜/s,q兲, 共23兲 pP⫽ kT F共˜/s,qp 兲 ⵜqF共p˜/s,q兲, 共24兲 s˙⫽s Q , 共25兲 ˙⫽ kT sF共p˜/s,q兲 ⵜp˜/sF共p˜/s,q兲⫺gkT⫺⌬HNose´. 共26兲
These formulations disrupt the separability of variables present in the original NH and NP equations of motion关Eqs.
共6兲–共9兲 and共13兲–共16兲, respectively兴. A time reversible dis-cretization of the GDD equations would involve the solution of nonlinear equations in q and p at every step. Iterative solution would require many evaluations of the potential en-ergy and its gradient at each step, likely adding tremendously to the computational burden. We address this issue by con-sidering two special classes of probability density functions that maintain variable separability:
Case 1: GDD for separable distribution functions Consider separable probability distribution functions of the form,
F共p,q兲⫽A共p兲B共q兲.
We can relate the separable density to the canonical density by way of effective kinetic and potential energies Keff and
Veffas F共p,q兲⫽e⫺Keffe⫺Veff, leading to Keff共p兲⫽⫺ 1 ln A共p兲, Veff共q兲⫽⫺ 1 ln B共q兲. 共27兲 Canonical sampling with respect to the effective Hamiltonian Heff⫽Keff⫹Veff is equivalent to sampling from the general-ized probability density F. Following the procedure outlined in the previous section, canonical sampling with Heffcan be achieved using the Nose´–Hoover GDD equations of motion, which for a separable distribution function are obtained as
q˙⫽ⵜpKeff共p兲, 共28兲 p˙⫽⫺ⵜqVeff共q兲⫺p, 共29兲 ˙⫽, 共30兲 ˙⫽1 Q关p Tⵜ pKeff共p兲⫺gkBT兴. 共31兲
Generation of the Nose´–Poincare´ equations of motion for this class of distributions follows similarly.
Note that these equations have a simple relationship with the NH Eqs.共6兲–共9兲. Any existing implementation of the NH
共or NP兲equations of motion can be easily modified for sepa-rable GDD by the replacement of M⫺1p byⵜKeff(p) in Eqs. 共6兲,共9兲, and V(q) by Veff(q) in Eq.共7兲.
The most important applications for GDD for separable distributions are those in which only the coordinate distribu-tion is altered through modificadistribu-tion of the potential. Such
potential-only modifications are at the heart of Voter dynamics18and the Tsallis statistics based methods for accel-erated sampling of Straub and Andricioaei.11 For such sys-tems Keff is equal to its standard form
1 2p
TM⫺1p and V eff is
given by Eq.共27兲. Implementation of GDD for such systems is straightforward as any existing Nose´–Hoover 共or Nose´– Poincare´兲 code could be used without modification 共other than the use of a modified input potential surface兲.
Case 2: GDD for distributions that are functions of the Hamiltonian
Here we consider distributions that are formal functions of the scalar Hamiltonian: F(H(p,q)). Defining the effective Hamiltonian as
Heff⫽共⫺1/兲ln F共H共p,q兲兲⬅f共H共p,q兲兲
with associated Nose´ Hamiltonian, the Nose´–Hoover GDD equations of motion for this case are
q˙⫽f
⬘
共H共p,q兲兲M⫺1p, 共32兲 p˙⫽⫺f⬘
共H共p,q兲兲ⵜV共q兲⫺p, 共33兲 ˙⫽, 共34兲 ˙⫽1 Q关f⬘
共H共p,q兲兲p TM⫺1p⫺gk BT兴. 共35兲We can arrive at the natural expression for the momenta by performing the time transformation dt/dˆ⫽1/f
⬘
(H(q,p)),dq/dˆ⫽M⫺1p, 共36兲 dp/dˆ⫽⫺ⵜV共q兲⫺ 1 f
⬘
共H共p,q兲兲p, 共37兲 d/dˆ⫽ 1 f⬘
共H共p,q兲兲, 共38兲 d/dˆ⫽ 1 Q冋
p TM⫺1p⫺ 1 f⬘
共H共p,q兲兲gkBT册
. 共39兲These equations have a suggestive form. The influence of the modified distribution is manifested solely in the thermostat variables. Deviation of f
⬘
(H) from unity can be viewed as a time-dependent scaling of simulation temperature T along with an inverse scaling of the thermostat mass Q.It is possible to rewrite these equations yet again to achieve separation of the coordinates and momenta. Just as the quantity in Eq.共10兲is constant along solutions of the NH equations, the GDD Eqs. 共36兲–共39兲 conserve the related quantity,
E0f⫽f共H共q,p兲兲⫹Q 2
2 ⫹gkBT. 共40兲 Making use of our assumption that F and 共and also f兲 is monotonic and hence one-to-one, we can solve Eq. 共40兲,
H共q,p兲⫽f⫺1
冉
E0f⫺Q2
2 ⫺gkBT
冊
共41兲 and define a new function of and,共,兲⫽
再
f⬘
冋
f⫺1冉
E0 f⫺Q 2 2 ⫺gkBT冊册冎
⫺1 共42兲so that the GDD equations become dq/dˆ⫽M⫺1p, 共43兲 dp/dˆ⫽⫺ⵜV共q兲⫺共,兲p, 共44兲 d/dˆ⫽共,兲, 共45兲 d/dˆ⫽ 1 Q关p TM⫺1p⫺共,兲gk BT兴. 共46兲
Equations 共43兲–共46兲 can be shown to conserve energy by differentiating Eq. 共40兲 and substituting Eqs.共41兲 and共42兲. These equations introduce coupling between the thermostat variables, but leave the coordinates and momenta separated, allowing for efficient discretization schemes. A similar trick can also be used to simplify the numerical calculations in the symplectic Nose´–Poincare´ method. We discuss numerical methods in the Appendix.
It must be pointed out that because the GDD equations for this full Hamiltonian case were derived with a time trans-formation dt/dˆ⫽(,) in Eq.共42兲, it is necessary to in-clude as a weighting function when computing the time average of any dynamical variable A(⌫⬅兵p,q,,其) using trajectories produced by Eqs. 共43兲–共46兲, since
具
A典
⬅1 T冕
0 T dtA关⌫共t兲兴⫽1 Tˆ冕
0 Tˆ dˆdt dˆA关⌫共ˆ兲兴 ⫽1 Tˆ冕
0 Tˆ dˆ共ˆ兲A关⌫共ˆ兲兴,where t and ˆ are the original and transformed time vari-ables, respectively, and Tˆ⫽兰0Tdt(dˆ /dt)⫽兰0Tdt(t). Given thatis only a function of the scalar variablesand, this is a rather trivial task.共For clarity, a derivation of the distri-butions generated by the equations of motion given in this section using the non-Hamiltonian statistical mechanics of Tuckerman et al.23is given in the Appendix.兲
The effect of the technique presented above is to intro-duce a formulation which is共1兲semiexplicit, and共2兲evolves dynamics in a modified time scale. The semiexplicit nature arises from the fact that a two-dimensional nonlinear system must be solved iteratively at each time step; however, since this formulation specifically does not require the computa-tion of forces or other system-dependent nonlinearities at each iteration, the solution of the small system can be opti-mized and should be regarded as roughly similar to comput-ing square roots or trigonometric functions 共which are also generally solved by simple iterative processes兲. For the com-putation of time correlation functions, the presence of the time transformation would cause some minor inconvenience since the trajectories would have to be transformed back to real time before averaging共but see Ref. 24 for an example of how this can be done efficiently using interpolation兲; how-ever, since time and, by extension, time correlation functions in these generalized density simulations have, at present, no physical interpretation, the presence of the time-transformation is unimportant.
IV. EXAMPLE APPLICATION: A VARIABLE TEMPERATURE DISTRIBUTION
The methods of this paper are very general in the sense that dynamical simulations can be made to sample any smooth, invertible density function F(p,q). In this section we propose a particular distribution function both for the purpose of demonstrating the methods of this paper and to outline a potentially useful method for accelerating sampling in systems with high barriers.
As mentioned earlier, one way to enhance the sampling of systems that are not ergodic on the time scale of standard simulation due to high barriers is to carry out the simulations at high temperature. The original distribution can the be re-covered by reweighting the trajectory to compensate for the change in the distribution. However, for most situations the low energy configurations that have large weight in the origi-nal distribution are not sampled with sufficient frequency at high temperatures to yield adequate statistics after reweight-ing.
To address the low energy sampling problem with high temperatures, we propose a generalized distribution that has the effect of raising temperature only in high energy regions while leaving the low energy dynamics unaffected. To begin, we define the monotonic function f␥(s), designed to smoothly switch between the identity function and a linear function of slope␥, f␥共s兲⫽
再
s if s⬍s0 as3⫹bs2⫹cs⫹d if s0⭐s⭐s1 ␦⫹␥共s⫺s1兲 if s⬎s1, 共47兲where s0⭐␦⭐s1 control the size of the switching window
and the shape of the switching function, and the polynomial coefficients of the switching function are given by
a⫽共2␦⫺共1⫺␥兲s0⫺共1⫹␥兲s1兲/共s0⫺s1兲3, b⫽共2共1⫺␥兲s02⫹共2⫹␥兲共s0s1⫹s1 2兲⫺3␦共s 0⫹s1兲兲/ 共s0⫺s1兲3, c⫽共␥s0共s0 2⫹ s0s1⫺2s1 2兲⫺ s1共4s0 2⫹ s0s1⫺6␦s0⫹s1 2兲兲 / 共s0⫺s1兲3, d⫽共s02␦共s0⫺3s1兲⫹s1共2s1⫹␥共s1⫺s0兲兲兲/共s0⫺s1兲3. We now define a probability distribution as a function the Hamiltonian H,
F共p,q兲⫽F␥共H共p,q兲兲⫽e⫺f␥共H兲/kT. 共48兲
The same switching function f␥ can be used to generate a separable distribution with
F共p,q兲⫽e⫺pTM⫺1p/2kTe⫺f␥共V共q兲兲/kT 共49兲
in which the momentum distribution remains canonical. At ␥⫽1 共and ␦⫽s1) both the Hamiltonian and potential
ver-sions of the variable temperature distribution reduce to the canonical distribution at temperature T. For␥⬍1, the distri-butions give canonical sampling in low energy 共total or po-tential, respectively兲regions (E⬍E0for some predetermined
value E0) of phase space at reference temperature T while This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
sampling high energy regions (E⬎E1) at the higher
tem-perature T(E)⫽T/␣(E), where ␣(E)⫽␥(1⫺(E1/E))
⫹(␦/E). Note that␣(E) approaches ␥in the limit of large E. This distribution modification has similarities with the Tsallis-based distributions used by Andricioaei and Straub11 and Plastino and Anteneodo10 in that the effective tempera-ture is a monotonically increasing function of energy; how-ever, in these cases the temperature共and thus the dynamics兲 is altered at all energies whereas in our present case the dynamics at low energies is unaltered.
V. NUMERICAL EXPERIMENTS
To test our methods, we consider the double well poten-tial
V共x兲⫽⑀共x4⫺2x2⫹1兲
with minima at x⫽⫾1 and barrier height⑀.
We have performed a number of GDD simulations using the variable temperature distribution 共48兲 in both its full Hamiltonian and potential forms. For all simulations, the ref-erence temperature was kT⫽⑀/10. For such low dimensional systems it is necessary to use Nose´–Hoover chains to en-hance the ergodicity of the dynamics共see the Appendix for discussion兲. In all simulations six thermostats were used. The switching window parameters in Eq. 共47兲 were taken as s0 ⫽3, s1⫽4, and␦⫽3.5共except when␥⫽1, then ␦⫽s1). We report experiments with the variable temperature parameter ␥⫽1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2. As ␥ decreases from unity, stability considerations dictate smaller time steps.
The time step was h⫽0.001 for the modified potential en-ergy calculations. For the full Hamiltonian approach, we use time step h⫽0.0001. As a demonstration of the accuracy of our new method for the full Hamiltonian simulation 共as measured by energy conservation兲, we show in Fig. 1 the relative energy error,兩E(t)⫺E(t⫽0)/E(t⫽0)兩, for a run of 5⫻106 time steps.
The left view of Fig. 2 shows the distribution of coordi-nates for the full Hamiltonian calculations at ␥⫽0.2. It can be seen by the good agreement with the theoretical distribu-tion that the coordinates along the trajectory are sampled according to F␥. Also shown is the reweighted distribution which recovers the canonical distribution. Note that at this temperature (kT⫽⑀/10) standard NH chain dynamics fails to sample effectively. As discussed earlier, a weighting factor of was used when computing averages to take care of the effects of the time transformation used to derive the GDD equations of motion. The right view of Fig. 2 shows the distribution of coordinates for the modified potential calcu-lations at␥⫽0.2. As for the results for the full Hamiltonian, it can be seen that the canonical coordinate distribution is also recovered by reweighting. Note that the unweighted co-ordinate distributions from the full Hamiltonian and modified potential formulations are not identical. In particular, the tra-jectory from the full Hamiltonian method spends slightly more time in high energy configurations.
In Fig. 3 we show the distribution of total energy along the computed full Hamiltonian trajectory for F␥with␥⫽0.2, which can be seen to closely approximate the theoretical en-ergy distribution for F␥. Also shown is the function f (H) from Eq. 共47兲, along with the computed values of f calcu-lated as the natural logarithm of the energy distribution from the trajectory.
In Fig. 4 we illustrate the success of the variable tem-perature density F␥ in hastening barrier crossings for the double well system. This figure shows waiting time plotted vs the temperature boost factor 1/␥. It can be seen that both the full Hamiltonian and modified potential approaches yield dramatic reductions in waiting time between barrier cross-ings.
VI. CONCLUSION
In this paper we have presented a general dynamical formalism, which we call Generalized Distribution
Dynam-FIG. 1. Relative energy error as a function of time for the full Hamiltonian GDD applied to the double well system discussed in the text.
FIG. 2. Left view: Full Hamiltonian simulation f␥with␥⫽0.2. Right view: Modified potential simulation with
␥⫽0.2. The solid curve gives the coor-dinate distribution of the computed trajectories, the dashed line gives the reweighting of the computed sampling to the canonical distribution, and the dotted lines give the theoretical ca-nonical distribution.
ics 共GDD兲, that generates trajectories that sample from any of a broad class of probability distribution functions. In ad-dition we show that the GDD scheme, which is based on the Nose´ thermostat,3can be easily implemented numerically for two classes of distribution functions: distributions that are functions of the full Hamiltonian and those that separate into a product of momentum and position distributions. In these two cases, the GDD scheme is equivalent to the dynamics of a system with a modified full Hamiltonian or effective po-tential energy surface, respectively. To implement GDD for these two classes, we outline specific numerical methods for both the Nose´–Hoover4 and Nose´–Poineare´5 real-time for-mulations of Nose´ dynamics. As an example, we have intro-duced a specific form of a probability density function, the variable temperature distribution, which has application in accelerating configurational sampling in systems with high energy barriers. To illustrate the numerical scheme and evaluate the method we performed numerical experiments using a one-dimensional bistable oscillator and demonstrate that the variable temperature distribution is very effective for accelerated sampling of coordinates when used in the effec-tive potential energy setting. We are currently applying this work to enhance the dynamical sampling of model polypep-tides.
ACKNOWLEDGMENTS
This work was undertaken during visits of E.J.B. and B.B.L. to the Center for Mathematical Modelling at the Uni-versity of Leicester, UK. Acknowledgment is made by E.J.B. to the donors of the Petroleum Research Fund, administered by the ACS, for partial support of this research. B.B.L. ac-knowledges the support of the National Science Foundation under Grant No. CHE-9970903. B.J.L. acknowledges the UK Engineering and Physical Sciences Research Council Grant No. GR/R03259/01. The authors also wish to thank Professor Mark Tuckerman for his careful reading and con-structive criticism of the manuscript.
APPENDIX: ALGORITHMIC DETAILS
1. Numerical methods for Nose´–Hoover GDD for distributions that are functions of the Hamiltonian
The NH Eqs.共6兲–共9兲can be discretized in a number of ways that generalize the basic Verlet approach. One such method is given in Appendix D of Ref. 25,
pn⫹1/2⫽pn⫺ ⌬t 2 共ⵜV共qn兲⫹n⫹1/2pn⫹1/2兲, 共A1兲 n⫹1/2⫽n⫹ ⌬t 2Q共pn⫹1/2 T M⫺1p n⫹1/2⫺gkBT兲, 共A2兲 n⫹1⫽n⫹⌬tn⫹1/2, 共A3兲 qn⫹1⫽qn⫹⌬tM⫺1pn⫹1/2, 共A4兲 n⫹1⫽n⫹1/2⫹ ⌬t 2Q共pn⫹1/2 T M⫺1pn⫹1/2⫺gkBT兲, 共A5兲 pn⫹1⫽pn⫹1/2⫺ ⌬t 2 共ⵜV共qn⫹1兲⫹n⫹1/2pn⫹1/2兲. 共A6兲 Equations 共A1兲 and 共A2兲 can be solved by computing the vector,
p ¯⫽pn⫺
⌬t
2 ⵜV共qn兲 and rewriting Eq.共A2兲,
FIG. 3. Left view: Total energy den-sity for the full Hamiltonian distribu-tion. Right view: The function f␥(H) reconstructed from trajectory energy sampling. The solid curve gives the to-tal energy distribution of the computed trajectories, the dashed line gives the reweighting of the computed sampling to the canonical distribution, and the dotted lines give the theoretical ca-nonical distribution.
FIG. 4. Sampling speedup measured as increased frequency of barrier cross-ings for the full Hamiltonian共stars兲and modified potential共circles兲methods using the variable temperature distribution. The high energy slope reduction factors is ␥⫺1. The speedup factor is the average waiting time between barrier crossings for each value of ␥, normalized by the average waiting time at␥⫽1.
n⫹1/2⫽n⫹ ⌬t 2Q
冉
p ¯TM⫺1p¯冉
1⫹⌬t 2 n⫹1/2冊
2⫺gkBT冊
.This scalar equation for n⫹1/2 can now be solved
analyti-cally or with an iterative solver. Using the computedn⫹1/2,
we can compute pn⫹1/2⫽ p ¯ 1⫹⌬t 2 n⫹1/2 .
The discretization scheme共A1兲–共A6兲can be generalized to the GDD Eqs. 共43兲–共46兲, pn⫹1/2⫽pn⫺ ⌬t 2 共ⵜV共qn兲⫹共n,n⫹1/2兲n⫹1/2pn⫹1/2兲, 共A7兲 n⫹1/2⫽n⫹ ⌬t 2Q共pn⫹1/2 T M⫺1pn⫹1/2⫺共n,n⫹1/2兲gkBT兲, 共A8兲 n⫹1/2⫽n⫹ ⌬t 2 共n,n⫹1/2兲n⫹1/2, 共A9兲 qn⫹1⫽qn⫹⌬tM⫺1pn⫹1/2, 共A10兲 n⫹1⫽n⫹1/2⫹ ⌬t 2 共n⫹1,n⫹1/2兲n⫹1/2, 共A11兲 n⫹1⫽n⫹1/2⫹ ⌬t 2Q共pn⫹1/2 T M⫺1p n⫹1/2 ⫺共n⫹1,n⫹1/2兲gkBT兲, 共A12兲 pn⫹1⫽pn⫹1/2⫺ ⌬t 2 共ⵜV共qn⫹1兲 ⫹共n⫹1,n⫹1/2兲n⫹1/2pn⫹1/2兲. 共A13兲
Two of the formulas are implicit. The procedure forn⫹1/2is
the same as for Eq. 共A2兲, while Eq.共A11兲 may require the use of an iterative method, depending on the nature of the function . In the numerical results presented here, we use Newton–Raphson iteration with tolerance of 10⫺12in double precision.
The GDD scaling function共,兲introduced in Eq.共42兲 serves two interesting purposes. The resulting dynamical for-malism retains the form of the NH equations, with coordi-nates and momenta coupled to a generalized thermostat共 sub-ject to a time transformation兲via the thermostat variables and . The introduction of allows for discretization schemes which are explicit in the coordinates and momenta, which is important for overall efficiency. Implementation of a timestepping scheme such as Eqs. 共A7兲–共A13兲 requires repeated evaluation of the function, requiring the inversion of the function f in Eq. 共17兲 which defines the effective Hamiltonian of the generalized density. In general an ana-lytic expression will not be available for f⫺1. For the work described here, we have implemented the GDD scaling func-tion using an algorithm which relies on the Newton– Raphson method for finding a zero of a scalar equation. For
the variable temperature distribution based on Eq. 共47兲, we evaluate (,)⫽1/f
⬘
( f⫺1(H)) by first evaluating ⌬E ⫽E0f⫺(Q2/2)⫺gkBT from Eq. 共42兲, then evaluatingf⫺1共⌬E兲 ⫽
再
⌬E if⌬E⬍H0 root of as3⫹bs2⫹cs⫹d⫺⌬E if H0⭐⌬E⭐H1 1 ␥共⌬E⫺␦兲⫹H1 if⌬E⬎H1, and f⬘
共f⫺1兲⫽再
1 if ⌬f⫺1⬍H0 3as2⫹2bs⫹c if H0⭐f⫺1⭐H1 ␥ if f⫺1⬎H1.With a good initial approximation 共such as⌬E or the aver-age 12(⌬E⫹(1/␥)(⌬E⫺␦)⫹H1)) the Newton–Raphson
method above converges to the desired root with two or three iterations in our experience, subject to a convergence toler-ance of 10⫺12in double precision calculations.
2. A symplectic numerical method for Nose´–Poincare´ GDD for distributions which are functions
of the Hamiltonian
Starting from the Nose´ extended Hamiltonian applied to the effective Hamiltonian,
HNose´f ⫽⫺f共H共q, p˜ /s兲兲⫹
2
2Q⫹gkBT ln s. 共A14兲 Assuming f is one to one, we can invert f to obtain, along the energy surface HNose´f ⫽E, the new Hamiltonian
H共q, p˜ /s兲⫺f⫺1
冉
E 0 f⫺2
2Q⫺gkT ln s
冊
⫽0. 共A15兲 The zero energy dynamics in this Hamiltonian correspond to HNose´⫽E0f dynamics. We next introduce a
time-transformation of the Poincare´-type, H→sH resulting in HNPf ⫽sH共q, p˜ /s兲⫺s f⫺1
冉
E0f⫺2
2Q⫺gkT ln s
冊
⫽0.共A16兲 It is natural to use a splitting method here, breaking the Hamiltonian into two parts according to the obvious additive decomposition and solving each term successively using an appropriate symplectic numerical method. Note that the splitting suggested here is different than that used recently by Nose´26in his variation of the Nose´–Poincare´ method, but the basic technique is similar. Integration of the term
H1⫽sH共q, p˜ /s兲 共A17兲
can be easily performed using the standard共and symplectic兲 Verlet method; note that during this fraction of the propaga-tion timestep, s will be constant. Formally, the integrapropaga-tion of
H2⫽⫺s f⫺1
冉
E0 f⫺ 2
2Q⫺gkT ln s
冊
共A18兲can be done analytically. However, this is relatively painful. A simpler approach is to use an implicit method, such as the implicit midpoint method. For a general Hamiltonian H(q, p), the midpoint method advances from step to step by solving
qn⫹1⫽qn⫹␦tⵜpH共qn⫹1/2,pn⫹1/2兲, 共A19兲 pn⫹1⫽pn⫹␦tⵜqH共qn⫹1/2,pn⫹1/2兲, 共A20兲
where qn⫹1/2⬅(qn⫹qn⫹1)/2 and pn⫹1/2 is defined similarly.
In the present case, this means solving a nonlinear system in
R2 at each time step.
3. Non-Hamiltonian statistical mechanics of the generalized distribution dynamics
Recently, Tuckerman et al.23 have developed a general procedure for the evaluation of the statistical distributions generated by non-Hamiltonian equations of motion. In this Appendix we utilize this formalism to confirm that the GDD equations of motion for distributions that are functions of the Hamiltonian关Eqs. 共36兲–共39兲and Eqs.共43兲–共46兲兴 do indeed generate the canonical distribution关or the canonical distribu-tion weighed by 共,兲 in the case of the time transformed equations of motion兴.
Consider a dynamical equation of motion,
x˙⫽G共x兲
with a conserved quantity E(x)⫽C, where C is a constant and x is the set of dynamical variables 共for a Hamiltonian system, these would be the coordinates and conjugate mo-menta兲. For this system one can define a divergence共 some-times referred to as a compressibility兲by
⬅ⵜi•x˙⫽
兺
ix˙i
xi.
The phase-space distribution is then given by23 共x兲⫽ 1
⍀e⫺w共x兲␦关E共x兲⫺C兴,
where⍀is the phase-space partition function共which serves to normalize the distribution兲 and w is defined such that w˙ ⫽. For a Hamiltonian system,⫽0 and the usual equation for the distribution is obtained.
For the untransformed Nose´–Hoover GDD equations of motion for distributions that are functions of the full Hamil-tonican关Eqs. 共32兲–共35兲兴, the divergence is
⫽ⵜp•p˙⫹ⵜq•q˙⫹ ˙ ⫹ ˙ ⫽f
⬙
共H共p,q兲兲关ⵜqH M⫺1p⫺ⵜpHⵜqV共q兲兴⫺g, ⫽⫺g⫽⫺g˙ ,where g is the number of degrees of freedom, the penulti-mate step follows sinceⵜpH⫽M⫺1p andⵜqH⫽ⵜqV(q), and the last step is a consequence of the equation of motion for
关Eq.共34兲兴. This equation forgives w⫽⫺g, which when combined with the conserved quantity关Eq.共41兲兴,
E0f⫽f共H共p,q兲兲⫹gkT⫹12Q 2
gives the phase space distribution 共p,q,,兲⫽ 1 ⍀e⫺g␦关f共H共p,q兲兲⫹gkT⫹12Q 2⫺E 0 f兴 . Integrating over the auxilliary variables and then gives the required distribution in the reduced phase space
共p,q兲⫽1
Zexp兵⫺f共H共p,q兲兲/kT其,
where Z is the corresponding partition function.
For the time-transformed equations关Eqs.共36兲–共39兲兴the analysis is similar, except that the divergence can be shown to be
⫽⫺ddˆ兵ln关f
⬘
共H兲兴⫹g其,which gives w⫽⫺ln关f
⬘
(H)兴⫹g 共the details are left as an exercise for the reader兲. The distribution is then共p,q,,兲⫽ 1 ⍀ e⫺g f
⬘
共H兲␦关f共H共p,q兲兲⫹gkT⫹ 1 2Q 2⫺E 0 f兴 . This differs from the distribution for the untransformed equa-tions derived above by a weighting factor 1/f⬘
(H) ⫽(,), so in the calculation of averages it is necessary to unbias the distribution by reweighting with a factor 1/, as discussed in the text. The final equations of motion for this system关Eqs.共43兲–共46兲兴that were derived by a rewriting of Eqs.共43兲–共46兲using Eqs.共42兲and共41兲can be easily shown to have a divergence of⫽⫺g˙ ,
which is identical to that for the original untransformed equations 关Eqs.共32兲–共35兲兴. This would seem to be a contra-diction in that the naive application of the non-Hamiltonian formulation given above would yield a distribution identical to that of the untransformed equations; however, the trajec-tory generated by Eqs. 共43兲–共46兲 is identical to that gener-ated by Eqs. 共36兲–共39兲and would therefore generate a dis-tribution weighted by . The resolution of this apparent paradox is to note that the conservation law for Eqs. 共43兲–
共46兲 is given by Eq. 共41兲 not by Eq. 共40兲 as used above. When the delta function in the above equation for the distri-bution is replaced with ␦关H(p,q)⫺f⫺1(E0f⫺(Q2/2) ⫺gkT)兴, and the nonlinear dependence of the delta func-tion on is taken into account in the intergration over the auxilliary variables, the desired distribution is recovered, as required.
4. Nose´–Hoover chains
For systems that are small or contain stiff oscillatory components, lack of ergodicity may render the Nose´ scheme ineffective. Chains of Nose´–Hoover thermostats have been shown22 to allow canonical sampling in these cases. For a chain of m⫹1 thermostat variables the equations of motion are
dq/d⫽M⫺1p, 共A21兲 dp/d⫽⫺ⵜV共q兲⫺0p, 共A22兲 d0/d⫽0, 共A23兲 d0/d⫽ 1 Q0关 pTM⫺1p⫺gkBT兴⫺10, 共A24兲 d1/d⫽1, 共A25兲 d1/d⫽ 1 Q1关 Q00 2⫺k BT兴⫺21, 共A26兲 ] dm⫺1/d⫽m⫺1, 共A27兲 dm⫺1/d⫽ 1 Qm⫺1关 Qm⫺2m⫺2 2 ⫺ kBT兴⫺mm⫺1, 共A28兲 dm/d⫽m, 共A29兲 dm/d⫽ 1 Qm关 Qm⫺1m2⫺1⫺kBT兴. 共A30兲 Along solutions of the Nose´–Hoover Chain共NHC兲equations the conserved quantity is
ENHC⫽p TM⫺1p 2 ⫹V共q兲⫹
兺
i⫽0 m Qii2 2 ⫹gkBT0 ⫹兺
i⫽1 m kBTi. 共A31兲Discretization schemes for NH chains are discussed in Refs. 22 and 27.
Extension of Nose´–Poincare´ to incorporate chains is somewhat delicate; this is work in progress by two of the authors.
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