bulgac-kusnezov-nosé-hoover thermostats · which molecular dynamics simulation at constant...

Bulgac-Kusnezov-Nosé-Hoover thermostats

Alessandro Sergi*School of Physics, University of KwaZulu-Natal, Pietermaritzburg, Private Bag X01 Scottsville, 3209 Pietermaritzburg, South Africa

Gregory S. Ezra†

Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA�Received 3 February 2010; published 19 March 2010�

In this paper, we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means ofnon-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined, one where theBulgac-Kusnezov demons are globally controlled by means of a single additional Nosé variable, and anotherwhere each demon is coupled to an independent Nosé-Hoover thermostat. Numerically stable and efficientmeasure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaoticproperties of the different phase space flows are numerically illustrated through the paradigmatic example ofthe one-dimensional harmonic oscillator. It is found that, while the simple Bulgac-Kusnezov thermostat isapparently not ergodic, both of the Nosé-Hoover controlled dynamics sample the canonical distributioncorrectly.

DOI: 10.1103/PhysRevE.81.036705 PACS number�s�: 05.10.�a, 05.20.Jj, 05.45.Pq, 65.20.De

I. INTRODUCTION

In condensed matter studies, there are many situations inwhich molecular dynamics simulation at constant tempera-ture �1–3� is needed. For example, this occurs when mag-netic systems are modeled in terms of classical spins �4–7�.Deterministic methods �8–10�, based on non-Hamiltoniandynamics �11–19�, can sample the canonical distribution pro-vided that the motion in the phase space of the relevant de-grees of freedom is ergodic �1,3�. However, classical spinsystems are usually formulated in terms of noncanonicalvariables �20,21�, without a kinetic energy expressed throughmomenta in phase space, so that Nosé dynamics cannot beapplied directly. To tackle this problem, Bulgac and Kusn-ezov �BK� introduced a deterministic constant-temperaturedynamics �22–24�, which can be applied to spins. A numberof numerical approaches to integration of spin dynamics canbe found in the literature �25–28�. However, BK dynamics,as any other deterministic canonical phase space flow, is ableto correctly sample the canonical distribution only if the mo-tion in phase space is ergodic on the timescale of the simu-lation. In general, this condition is very difficult to check forstatistical systems with many degrees of freedom, while it isknown that, despite its simplicity, the one-dimensional har-monic oscillator provides a difficult and important challengefor deterministic thermostatting methods �9,29–31�.

In this paper, we accomplish two goals. First, by reformu-lating BK dynamics through non-Hamiltonian brackets�14,15� in phase space, we introduce two generalized ver-sions of the BK time evolution which are able to sample thecanonical distribution for a stiff harmonic system. Second,using a recently introduced approach based on the geometryof non-Hamiltonian phase space �19�, we are able to derivestable and efficient measure-preserving and time-reversiblealgorithms in a systematic way for all the phase space flowstreated here.

The BK phase space flow introduces temperature controlby means of fictitious coordinates �and their associated mo-menta in an extended phase space� traditionally called “de-mons.” Our generalizations of the BK dynamics are obtainedby controlling the BK demons themselves by means of ad-ditional Nosé-type variables �8�. In one case, the BK demonsare controlled globally by means of a single additional Nosé-Hoover thermostat �8,9�. In the following this will be re-ferred to as Bulgac-Kusnezov-Nosé-Hoover �BKNH� dy-namics. In the second case, each demon is coupled to anindependent Nosé-Hoover thermostat. This will be called theBulgac-Kusnezov-Nosé-Hoover chain �BKNHC�, and corre-sponds to “massive” NH thermostatting of the demon vari-ables �32�. The ability to derive numerically stable measure-preserving time-reversible algorithms �19� for Nosécontrolled BK dynamics is very encouraging for future ap-plications to thermostatted spin systems.

This paper is organized as follows. In Sec. II, we brieflysketch the unified formalism for non-Hamiltonian phasespace flows and measure-preserving integration. The BKdynamics is formulated in phase space and a measure-preserving integration algorithm is derived in Sec. III.The BKNH and BKNH-chain thermostats are treated inSecs. IV and V, respectively. Numerical results for the one-dimensional harmonic oscillator using these thermostats arepresented and discussed in Sec. VI. Section VII reports ourconclusions.

In addition we include several appendices. A useful op-erator formula is derived in Appendix A, while invariantmeasures for the BK, BKNH, and BKNHC phase spaceflows are derived in Appendices B–D, respectively.

II. NON-HAMILTONIAN BRACKETS AND MEASURE-PRESERVING ALGORITHMS

Consider an arbitrary system admitting a time-independent �extended� Hamiltonian expressed in terms ofthe phase space coordinates xi, i=1, . . . ,2N. In this case, theHamiltonian can be interpreted as the conserved energy ofthe system.

*[email protected]†[email protected]

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http://dx.doi.org/10.1103/PhysRevE.81.036705

Upon introducing an antisymmetric tensor field �general-ized Poisson tensor �21,33�� in phase space, B�x�=−BT�x�,one can define non-Hamiltonian brackets �14–16� as

�a,b� = �i,j=1

2n�a

�xiBij

�b

�xj, �1�

where a=a�x� and b=b�x� are two arbitrary phase spacefunctions. The bracket defined in Eq. �1� is classified as non-Hamiltonian �14–16� since, in general, it does not obey theJacobi relation, i.e., in general the Jacobiator J�0, where�21�

J = �a,�b,c�� + �b,�c,a�� + �c,�a,b�� , �2�

with c=c�x� arbitrary phase space function �in addition tothe functions a and b, previously introduced�. If J�0, thetensor Bij is said to define an “almost-Poisson” structure �34��such systems have also been called “pseudo-Hamiltonian”�33��.

An energy conserving and in general non-Hamiltonianphase space flow is then defined by the vector field

xi = �xi,H� = �j=1

2N

Bij�H

�xj, �3�

where conservation of H�x� follows directly from the anti-symmetry of Bij.

It has previously been shown how equilibrium statisticalmechanics can be comprehensively formulated within thisframework �16�. It is also possible to recast the above for-malism and the corresponding statistical mechanics in thelanguage of differential forms �17,18�. If the matrix B isinvertible �this is true for all the cases considered here�, withinverse �ij, we can define the 2-form �35�

� =1

2�ijdxi ∧ dxj . �4�

The dynamics of Eq. �3� is then Hamiltonian if and only ifthe form Eq. �4� is closed, i.e., has zero exterior derivative,d�=0 �35�. This condition is independent of the particularsystem of coordinates used to describe the dynamics.

The structure of Eq. �3� can be taken as the starting pointfor derivation of efficient time-reversible integration algo-rithms that also preserve the appropriate measure in phasespace �19�. Measure-preserving algorithms can be derivedupon introducing a splitting of the Hamiltonian

H = ��=1

ns

H�, �5�

which in turn induces a splitting of the Liouville operatorassociated with the non-Hamiltonian bracket in Eq. �1�,

L�xi = �xi,H�� = �j=1

2N

Bij�H�

�xj. �6�

When the phase space flow has a nonzero compressibility

� = �i,j=1

2N�Bij

�xi

�H

�xj, �7�

the statistical mechanics must be formulated in terms of amodified phase space measure �12–18�

� = e−w�x�� 8�

where

� = dx1 ∧ dx2 ∧ . . . ∧ dx2N �9�

is the standard phase space volume element �volume form�35�� and the statistical weight w�x� is defined by

dw

dt= ��x� . �10�

It has been shown that, provided the condition

�

�xj�e−w�x�Bij� = 0, i = 1, . . . 2N �11�

is satisfied, then

L�� = 0 for every � , �12�

so that the volume element � is invariant under each of theL� �19�. The condition �11� is satisfied for all the cases con-sidered below, so that, exploiting the decomposition in Eq.�6�, algorithms derived by means of a symmetric Trotter fac-torization of the Liouville propagator:

exp��L� = ��=1

ns−1

exp �

2L�exp exp��Lnx

� ��=1

ns−1

exp �

2Lns−�

�13�

are not only time-reversible but also measure preserving.

III. PHASE SPACE FORMULATIONOF THE BK THERMOSTAT

A phase space formulation of the BK thermostat can beachieved upon introducing the Hamiltonian

HBK =p2

2m+ V�q� +

K1�p��m�

+K2�p�

m

+ kBT�� + � �14a�

=H�q,p� +K1�p��

m�

+K2�p�

m

+ kBT�� + � , �14b�

where �q , p� are the physical degrees of freedom �coordinatesand momenta�, with mass m, to be simulated at constanttemperature T, while � and are the BK ‘demons’, withcorresponding inertial parameters m� and m, and associatedmomenta �p� , p� �22–24�. K1 and K2 provide the kinetic en-ergy of demon variables, and for the moment are left arbi-trary.

Upon defining the phase space point as x= �q ,� , , p , p� , p�= �x1 ,x2 ,x3 ,x4 ,x5 ,x6�, one can introducean antisymmetric BK tensor field as

ALESSANDRO SERGI AND GREGORY S. EZRA PHYSICAL REVIEW E 81, 036705 �2010�

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BBK = �0 0 0 1 0 − G2

0 0 0 0�G1

�p0

0 0 0 0 0�G2

�q

− 1 0 0 0 − G1 0

0 −�G1

�p0 G1 0 0

G2 0 −�G2

�q0 0 0

� , �15�

where G1 and G2 are functions of system variables �p ,q�only.

Substituting BBK and HBK into Eq. �3�, we obtain theenergy-conserving equations

q =�H

�p−

G2�q,p�m

�K2

�p

, �16a�

� =1

m�

�G1

�p

�K1

�p�

, �16b�

=1

m

�G2

�q

�K2

�p

, �16c�

p = −�H

�q−

G1�q,p�m�

�K1

�p�

, �16d�

p� = G1�H

�p− kBT

�G1

�p, �16e�

p = G2�H

�q− kBT

�G2

�q. �16f�

The associated invariant measure for the BK flow is dis-cussed in Appendix B.

Algorithm for BK dynamics

In order to derive a measure preserving algorithms, thefirst step, following Eq. �5�, is to introduce a splitting of HBK:

H1BK = V�q� , �17a�

H2BK =

p2

2m, �17b�

H3BK = kBT� , �17c�

H4BK = kBT , �17d�

H5BK =

K1�p��m�

, �17e�

H6BK =

K2�p�m

. �17f�

A measure-preserving splitting of the Liouville operator thenfollows from Eq. �6�:

L1BK = −

�V

�q

�

�p+ G2

�V

�q

�

�p

, �18a�

L2BK =

p

m

�

�q+ G1

p

m

�

�p�

, �18b�

L3BK = − kBT

�G1

�p

�

�p�

, �18c�

L4BK = − kBT

�G2

�q

�

�p

, �18d�

L5BK =

1

m�

�G1

�p

�K1

�p�

�

��−

G1

m�

�K1

�p�

�

�p, �18e�

L6BK = −

G2

m

�K2

�p

�

�q+

1

m

�G2

�q

�K2

�p

�

�. �18f�

Upon choosing a symmetric Trotter factorization of the BKLiouville operator based on the decomposition

LBK = ��=1

8

L�BK �19�

a measure-preserving algorithm can be produced in full gen-erality.

In practice, a choice of K1, K2, G1, and G2 must be madein order obtain explicit formulas. In this paper, we make thefollowing simple choices:

G1 = p , �20a�

G2 = q , �20b�

K1 =p�

2

2, �20c�

K2 =p

2

2. �20d�

In terms of Eqs. �20a�–�20d�, the antisymmetric BK tensorbecomes

BBK = �0 0 0 1 0 − q

0 0 0 0 1 0

0 0 0 0 0 1

− 1 0 0 0 − p 0

0 − 1 0 p 0 0

q 0 − 1 0 0 0

� , �21�

and the Hamiltonian reads

HBK = H�q,p� +p�

2

2m�

+p

2

2m

+ kBT�� + � . �22�

BULGAC-KUSNEZOV-NOSé-HOOVER THERMOSTATS PHYSICAL REVIEW E 81, 036705 �2010�

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The split Liouville operators now simplify as follows:

L1BK = −

�V

�q

�

�p+ q

�V

�q

�

�p

, �23a�

L2BK =

p

m

�

�q+

p2

m

�

�p�

, �23b�

L3BK = − kBT

�

�p�

, �23c�

L4BK = − kBT

�

�p

, �23d�

L5BK =

p�

m�

�

��−

p�

m�

p�

�p+

p�2

m�

�

�p

, �23e�

L6BK = −

p

m

q�

�q+

p

m

�

�+

p2

m

�

�p�

. �23f�

For the purposes of defining an efficient integration algo-rithm, we combine commuting Liouville operators as fol-lows:

LABK L1

BK + L4BK = F�q�

�

�p+ Fp

�

�p

, �24a�

LBBK L2

BK + L3BK =

p

m

�

�q+ Fp�

�

�p�

, �24b�

LCBK L5

BK + L6BK = −

p�

m�

p�

�p−

p

m

q�

�q+

p�

m�

�

��+

p

m

�

�,

�24c�

where

F�q� = − �V/�q , �25a�

Fp= q

�V

�q− kBT , �25b�

Fp�=

p2

m− kBT . �25c�

Defining

U�BK�� = exp��L�

BK� , �26�

where �=A ,B ,C, one possible reversible measure-preserving integration algorithm for the BK thermostat isthen

U��BK = UBBK� �

4�UC

BK� �

2�UB

BK� �

4�UA

BK��

� UBBK� �

4�UC

BK� �

2�UB

BK� �

4� . �27�

Using the so-called direct translation technique �36� wecan expand the above symmetric breakup of the Liouvilleoperator into a pseudocode form, ready to be implementedon the computer:

�i� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BK� �

4� ,

�ii��p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

�:UCBK� �

2� ,

�iii� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BK� �

4� ,

�iv� �p → p + �F

p → p + �Fp

�:UABK�� ,

�v��q → q +

�

4

p

m

→ +�

4

p

m

p� → p� +�

4Fp�

�:UBBK� �

4� ,

�vi��p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

�:UCBK� �

2� ,

�vii� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BK� �

4� .


036705-4

IV. BULGAC-KUSNEZOV-NOSÉ-HOOVER DYNAMICS

The BKNH Hamiltonian

HBKNH = H�q,p� +K1�p��

m�

+K2�p�

m

+p

2

2m

+ kBT�� + � + 2kBT �28�

is simply the BK Hamiltonian augmented by the Nosé vari-ables � , p� with mass m. With the antisymmetric BKNHtensor

ℬBKNH

=�0 0 0 0 1 0 − G2 0

0 0 0 0 0�G1

�p0 0

0 0 0 0 0 0�G2

�q0

0 0 0 0 0 0 0 1

− 1 0 0 0 0 − G1 0 0

0 −�G1

�p0 0 G1 0 0 − p�

G2 0 −�G2

�q0 0 0 0 − p

0 0 0 − 1 0 p� p 0

� ,

�29�

we obtain from Eq. �3� equations of motion forthe phase space variables x= �q ,� , , , p , p� , p , p�= �x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8�:

q =�H

�p−

G2�q,p�m

�K2

�p

, �30a�

� =1

m�

�G1

�p

�K1

�p�

, �30b�

=1

m

�G2

�q

�K2

�p

, �30c�

=p

m

, �30d�

p = −�H

�q−

G1�q,p�m�

�K1

�p�

, �30e�

p� = G1�H

�p− kBT

�G1

�p− p�

p

m

, �30f�

p = G2�H

�q− kBT

�G2

�q− p

p

m

, �30g�

p =p�

m�

�K1

�p�

+p

m

�K2

�p

− 2kBT . �30h�

Here, a single Nosé variable is coupled to both of the BKdemons � and . The associated invariant measure is dis-cussed in Appendix C.

Algorithm for BKNH dynamics

The Hamiltonian can be split as

H1BKNH = V�q� , �31a�

H2BKNH =

p2

2m, �31b�

H3BKNH = kBT� , �31c�

H4BKNH = kBT , �31d�

H5BKNH =

K1�p��m�

, �31e�

H6BKNH =

K2�p�m

, �31f�

H7BKNH =

p2

2m

, �31g�

H8BKNH = 2kBT , �31h�

The measure-preserving splitting �19� of the Liouville opera-tor

L� = BijBKNH�H�

BKNH

�xj

�

�xi, �32�

yields

L1BKNH = −

�V

�q

�

�p+ G2

�V

�q

�

�p

, �33a�

L2BKNH =

p

m

�

�q+ G1

p

m

�

�p�

, �33b�

L3BKNH = − kBT

�G1

�p

�

�p�

, �33c�

L4BKNH = − kBT

�G2

�q

�

�p

, �33d�

L5BKNH =

1

m�

�G1

�p

�K1

�p�

�

��−

G1

m�

�K1

�p�

�

�p+

p�

m�

�K1

�p�

�

�p

,

�33e�

L6BKNH = −

G2

m

�K2

�p

�

�q+

1

m

�G2

�q

�K2

�p

�

�+

p

m

�K2

�p

�

�p

,

�33f�

L7BKNH =

p

m

�

�−

p

m

p�

�

�p�

−p

m

p

�

�p

, �33g�

L8BKNH = − 2kBT

�

�p

. �33h�


036705-5

At this stage, we leave the general formulation and adoptthe particular choice of K1, K2, G1, and G2 given in Eq. �20�.The antisymmetric BKNH tensor becomes

BBKNH = �0 0 0 0 1 0 − q 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

− 1 0 0 0 0 − p 0 0

0 − 1 0 0 p 0 0 − p�

q 0 − 1 0 0 0 0 − p

0 0 0 − 1 0 p� p 0

� ,

�34�

and the Hamiltonian simplifies to

HBKNH = H�q,p� +p�

2

2m�

+p

2

2m

+p

2

2m

+ kBT�� + � + 2BT .

�35�

The split Liouville operators are now

L1BKNH = −

�V

�q

�

�p+ q

�V

�q

�

�p

, �36a�

L2BKNH =

p

m

�

�q+

p2

m

�

�p�

, �36b�

L3BKNH = − kBT

�

�p�

, �36c�

L4BKNH = − kBT

�

�p

, �36d�

L5BKNH =

p�

m�

�

��−

p�

m�

p�

�p+

p�2

m�

�

�p

, �36e�

L6BKNH = −

p

m

q�

�q+

p

m

�

�+

p2

m

�

�p

, �36f�

L7BKNH =

p

m

�

�−

p

m

p�

�

�p�

−p

m

p

�

�p

, �36g�

L8BKNH = − 2kBT

�

�p

. �36h�

For the purposes of defining an efficient integration algo-rithm, we combine commuting Liouville operators as fol-lows:

LABKNH L1

BKNH + L4BKNH + L7

BKNH

= F�q��

�p+

p

m

�

�−

p

m

p�

�

�p�

+ �−p�

m�

p + Fp� �

�p

�37a�

LBBKNH L2

BKNH + L3BKNH =

p

m

�

�q+ Fp�

�

�p�

�37b�

LCBKNH L5

BKNH + L6BKNH + L8

BKNH

= −p�

m�

p�

�p−

p

m

q�

�q+

p�

m�

�

��+

p

m

�

�+ Fp

�

p

,

�37c�

where

F�q� = −�V

�q, �38a�

Fp= q

�V

�q− kBT , �38b�

Fp�=

p2

m− kBT , �38c�

Fp=

p�2

m�

+p

2

m

− 2kBT . �38d�

In LA there appears an operator with the form

Li = �−pk

mkpi + Fpi

� �

�pi, �39�

where �k , i�= �� ,� for LA. The action of the propagator as-sociated with this operator on pi is derived in Appendix A,and is given by

e�Lipi = pie−��pk/mk� + �Fpi

e−��pk/2mk��pk

2mk�−1

sinh�pk

2mk .

�40�

The apparently singular function

��pk

2mk�−1

sinh�pk

2mk �41�

is in fact well behaved as pk→0, and can be expanded in aMaclaurin series to suitably high order �37�. In our imple-mentation we used an eighth order expansion.

The propagators for the BKNH dynamics can now be de-fined as

U�BKNH�� = exp��L�

BKNH� , �42�

where �=A ,B ,C. One possible reversible measure-preserving integration algorithm for the BKNH thermostatcan then be derived from the following Trotter factorization:

U��BKNH = UBBKNH� �

4�UC

BKNH� �

2�UB

BKNH� �

4�UA

BKNH��

� UBBKNH� �

4�UC

BKNH� �

2�UB

BKNH� �

4� . �43�

The direct translation technique gives the followingpseudocode:


036705-6

�i��q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BKNH� �

4� ,

�ii� �p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

p → p +�

2Fp�

�:UCBKNH� �

2� ,

�iii� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BKNH� �

4� ,

�iv��p → p + �F�q�

p → p + �Fp

→ + �p

m

p� → p� exp− �p

m�:UA

BKNH�� ,

�v� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BKNH� �

4� ,

�vi� �p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

p → p +�

2Fp

�:UCBKNH� �

2� ,

�vii� �q → q +�

4

p

m

p� → p� +�

4Fp��:UB

BKNH� �

4� .

V. BULGAC-KUSNEZOV-NOSÉ-HOOVER CHAIN

For simplicity, we explicitly treat only the case, in whichthe p� and p demons are each coupled to a standard Nosé-Hoover thermostat �length one�. It would be straightforwardto couple each of the demons to NH chains �32�, and thegeneral case can be easily inferred from what follows. Definethe Hamiltonian

HBKNHC = H�q,p� +K1�p��

m�

+K2�p�

m

+p

2

2m

+p�

2

2m�

+ kBT�� + + + �� . �44�

Upon defining the phase space point x= �q ,� , , ,� , p ,p� , p , p , p��= �x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10� and theantisymmetric BKNHC tensor

BBKNHC =�0 0 0 0 0 1 0 − G2 0 0

0 0 0 0 0 0�G1

�p0 0 0

0 0 0 0 0 0 0�G2

�q0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

− 1 0 0 0 0 0 − G1 0 0 0

0 −�G1

�p0 0 0 G1 0 0 − p� 0

G2 0 −�G2

�q0 0 0 0 0 0 − p

0 0 0 − 1 0 0 p� 0 0 0

0 0 0 0 − 1 0 0 p 0 0

� , �45�


036705-7

associated non-Hamiltonian equations of motion are

xi = BijBKNHC�HBKNHC

�xj�46�

with i=1, . . . ,10.

Algorithm for BKNHC chain dynamics

Splitting the BKNHC chain Hamiltonian as

H1BKNHC = V�q� , �47a�

H2BKNHC =

p2

2m, �47b�

H3BKNHC = kBT� , �47c�

H4BKNHC = kBT , �47d�

H5BKNHC =

K1�p��m�

, �47e�

H6BKNHC =

K2�p�m

, �47f�

H7BKNHC =

p2

2m

, �47g�

H8BKNHC = kBT , �47h�

H9BKNHC =

p�2

2m�

, �47i�

H10BKNHC = kBT� , �47j�

we obtain the corresponding measure-preserving splitting ofthe Liouville operator

L� = BijBKNHC�H�

BKNHC

�xj

�

�xi. �48�

At this stage we go directly to Eq. �20�. The antisymmet-ric Nosé-Hoover-Bulgac-Kusnezov tensor becomes

BBKNHC

= �0 0 0 0 0 1 0 − q 0 0

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

− 1 0 0 0 0 0 − p 0 0 0

0 − 1 0 0 0 p 0 0 − p� 0

q 0 − 1 0 0 0 0 0 0 − p

0 0 0 − 1 0 0 p� 0 0 0

0 0 0 0 − 1 0 0 p 0 0

� ,

�49�

the Hamiltonian

HBKNHC = H�q,p� +p�

2

2m�

+p

2

2m

+p

2

2m

+p�

2

2m�

+ kBT�� + + + �� 50�

and associated Liouville operators

L1BKNHC = −

�V

�q

�

�p+ q

�V

�q

�

�p

, �51a�

L2BKNHC =

p

m

�

�q+

p2

m

�

�p�

, �51b�

L3BKNHC = − kBT

�

�p�

, �51c�

L4BKNHC = − kBT

�

�p

, �51d�

L5BKNHC =

p�

m�

�

��−

p�

m�

p�

�p+

p�2

m�

�

�p

, �51e�

L6BKNHC = −

p

m

q�

�q+

p

m

�

�+

p2

m

�

�p�

, �51f�

L7BKNHC =

p

m

�

�−

p

m

p�

�

�p�

, �51g�

L8BKNHC = − kBT

�

�p

, �51h�

L9BKNHC =

p�

m�

�

��−

p�

m�

p

�

�p

, �51i�

L10BKNHC = − kBT

�

�p�

. �51j�

We combine commuting Liouville operators as follows:

LABKNHC L1

BKNHC + L4BKNHC + L9

BKNHC

= F�q��

�p+

p�

m�

�

��+ �−

p�

m�

p + Fp� �

�p

, �52a�


036705-8

LBBKNHC L2


BKNHC

=p

m

�

�q+

p

m

�

�+ �−

p

m

p� + Fp��

�p�

, �52b�

LCBKNHC L5


BKNHC + L10BKNHC

= −p�

m�

p�

�p−

p

m

q�

�q+

p�

m�

�

��

+p

m

�

�+ Fp

�

p

+ Fp�

�

p�

, �52c�

whereF�q� = −

�V

�q, �53a�

Fp= q

�V

�q− kBT , �53b�

Fp�=

p2

m− kBT , �53c�

Fp=

p�2

m�

− kBT , �53d�

Fp�=

p2

m

− kBT . �53e�

Both in LABKNHC and LB

BKNHC there appears an operator withthe form

Li = �−pk

mkpi + Fpi

� �

�pi, �54�

where �k , i�= �� ,� for LA and �k , i�= � ,�� for LB. Againfollowing the derivation in Appendix A, we find

e�Lipi = pie−��pk/mk� + �Fpi

e−��pk/2mk��pk

2mk�−1

sinh�pk

2mk .

�55�

The function ��pk

2mk�−1sinh��

pk

2mk� is treated through an eighth

order expansion �37�.The propagators

U�BKNHC�� = exp��L�

BKNHC� , �56�

with �=A ,B ,C can now be introduced. One possible revers-ible measure-preserving integration algorithm for theBKNHC chain thermostat is then

U��BKNHC = UBBKNHC� �

4�UC

BKNHC� �

2�UB

BKNHC� �

4�

� UABKNHC��

� UBBKNHC� �

4�UC

BKNHC� �

2�UB

BKNHC� �

4� .

�57�

In pseudocode form, we have the resulting integration algo-rithm:

�i��q → q +

�

4

p

m

→ +�

4

p

m

p� → p�e−��/4��p/m� +

�

4Fp�

e−��/4��p/2m��

4

p

2m�−1

sinh �

4

p

2m�:UB

BKNHC� �

4� ,

�ii��p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

p → p +�

2Fp�

p� → p� +�

2Fp�

�:UCBKNHC� �

2� ,


036705-9

�iii��q → q +

�

4

p

m

→ +�

4

p

m

p� → p�e−��/4��p/m� +

�

4Fp�

e−��/4��p/2m��

4

p

2m�−1

sinh �

4

p

2m�:UB

BKNHC� �

4� ,

�iv� �p → p + �F

� → � + �p�

m�

p → pe−��p�/m�� + �Fp

e−��p�/2m��p�

2m��−1

sinh�p�

2m��:UA

BKNHC�� ,

�v��q → q +

�

4

p

m

→ +�

4

p

m

p� → p�e−��/4��p/m� +

�

4Fp�

e−��/4��p/2m��

4

p

2m�−1

sinh �

4

p

2m�:UB

BKNHC� �

4� ,

�vi��p → p exp−

�

2

p�

m�

q → q exp−�

2

p

m

� → � +�

2

p�

m�

→ +�

2

p

m

p → p +�

2Fp

p� → p� +�

2Fp�

�:UCBKNHC� �

2� ,

�vii��q → q +

�

4

p

m

→ +�

4

p

m

p� → p�e−��/4��p/m� +

�

4Fp�

e−��/4��p/2m��

4

p

2m�−1

sinh �

4

p

2m�:UB

BKNHC� �

4� .

VI. NUMERICAL RESULTS

In its simplicity, the dynamics of a harmonic mode in onedimension is a paradigmatic example for checking the cha-otic �ergodic� properties of constant-temperature phase spaceflows and the correct sampling of the canonical distribution.It is well known that it is necessary to generalize basic Nosé-

Hoover dynamics �1,8,9� to thermostats such as the Nosé-Hoover chain �32,37� in order to produce correct constant-temperature averages for systems such as the harmonic os-cillator.

Some time ago, BK dynamics was devised to provide adeterministic thermostat for systems such as classical spins


036705-10

�23,24�. To ensure efficient thermostatting, BK found it nec-essary to introduce several demons per thermostatted degreeof freedom, where each demon was taken to have a differentand in principle complicated coupling to the system degreeof freedom �23,24�. In the present work, we keep the form ofthe system-thermostat coupling as simple as possible, in or-der to facilitate the formulation of explicit, reversible andmeasure-preserving integrators �19�. It is then of interest toinvestigate the ability of our BK-type thermostats to producethe correct canonical sampling in the case of the harmonicoscillator. Interest in harmonic modes is also justified by thepossibility of devising models of condensed matter systemsin terms of coupled spins and harmonic modes, as alreadydone in quantum dynamics with so-called spin-boson models�38�. We therefore investigate the performance of our inte-gration schemes on the simple one-dimensional harmonic os-cillator.

For the particular calculations reported here, the oscillatorangular frequency, all masses and kBT were taken to be unity.The time step in all cases was �=0.0025, and all runs werecalculated for 106 time steps, starting from the same initialconditions: harmonic oscillator coordinate q=0.3, all otherphase space variables zero at t=0.

The measure-preserving algorithms derived here result instable numerical integration for all the three cases treated:BK, BKNH, and BKNHC chain dynamics. Figure 1 showsthe three extended Hamiltonians �normalized by their respec-tive initial time value� versus time. All three Hamiltoniansare numerically conserved by the corresponding measure-preserving algorithm to very high accuracy �which is main-tained in all the three cases�.

However, the basic BK phase space flow is not capable ofproducing the correct canonical sampling for a harmonicmode. This can be easily checked since the canonical distri-bution function of the harmonic oscillator is isotropic inphase space and its radial dependence can be calculated ex-actly. Details of this way of visualizing the phase space sam-pling have already been given in �14,15�. Figure 2, display-ing the comparison between the theoretical and thecalculated value of the radial probability in phase space,

clearly shows that the BK dynamics is not able to producecanonical sampling. A look at the inset of Fig. 2, showing thephase space distribution for the harmonic mode, also imme-diately shows that the dynamics is not ergodic.

The same analysis has been carried out for BKNH andBKNHC phase space flows, and these are displayed in Figs.3 and 4, respectively. Within numerical errors, both BKNHand BKNHC thermostats are able to produce the correct ca-nonical distribution for the stiff harmonic modes.

Introduction of a single, global Nosé-type variable in theBKNH thermostat effectively introduces additional couplingbetween the two demon variables. The effectiveness of theBKNH thermostat is consistent with our findings �results notdiscussed here� that introduction of explicit coupling be-tween demons in BK thermostat dynamics also leads to effi-cient thermostatting of the harmonic oscillator.

VII. CONCLUSIONS

We have formulated Bulgac-Kusnezov �23,24�, Nosé-Hoover controlled Bulgac-Kusnezov, and Bulgac-Kusnezov-Nosé-Hoover chain thermostats in phase space by means of

1

1.01

1.02

1.03

1.04

0 500 1000 1500 2000 2500

H

t

HBK

HBKNH

HBKNHC

FIG. 1. Comparison of the total extended Hamiltonian versustime �normalized with respect to its value at t=0� for the harmonicoscillator undergoing simple Bulgac-Kusnezov dynamics �HBK�,NH controlled Bulgac-Kusnezov dynamics �HBKNH�, and Bulgac-Kusnezov-Nosé-Hoover chain dynamics �HBKNHC�. Two curveshave been displaced vertically for clarity. The time-reversiblemeasure-preserving algorithms developed in this paper conserve theextended Hamiltonian to high accuracy in all three cases.

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.5 4

P

r

-4-3-2-101234

-4-3-2-1 0 1 2 3 4

p

q

FIG. 2. Radial phase space probability for a harmonic oscillatorunder Bulgac-Kusnezov dynamics. The continuous line shows thetheoretical value while the black bullets display the numerical re-sults. The inset displays a plot of the phase space distribution ofpoints along the single trajectory used to compute the radialprobability.

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.5 4

P

r

-4-3-2-101234

-4-3-2-1 0 1 2 3 4

p

q

FIG. 3. Radial phase space probability for a harmonic oscillatorunder Nosé-Hoover controlled Bulgac-Kusnezov dynamics. Thecontinuous line shows the theoretical value while the black bulletsdisplay the numerical results. The inset displays a plot of the phasespace distribution of points along the single trajectory used to com-pute the radial probability.


036705-11

non-Hamiltonian brackets �14,15�. We have derived time-reversible measure-preserving algorithms �19� for these threecases and showed that additional control by a single Nosé-Hoover thermostat or independent Nosé-Hoover thermostatsis necessary to produce the correct canonical distribution fora stiff harmonic mode.

Measure-preserving dynamics of the kind discussed hereis associated with equilibrium simulations �where, for ex-ample, there is a single temperature parameter T�. Stationaryphase space distributions associated with nonequilibriumsituations are much more complicated than the smooth equi-librium densities analyzed in the present paper �11,39,40�.Nonequilibrium simulations of heat flow could be carried outby extending the present approach to multimode systems�e.g., a chain of oscillators� coupled to BK-type demons withassociated NH thermostats corresponding to two differenttemperatures �41–43�.

The techniques presented here for derivation and imple-mentation of thermostats have been shown to be efficient andversatile. We anticipate that analogous approaches can beusefully applied to systems of classical spins coupled to bothharmonic and anharmonic modes.

APPENDIX A: OPERATOR FORMULA

We wish to determine the action of the propagator asso-ciated with the Liouville operator Eq. �39�. This is equivalentto solving the evolution equation �recall i�k�

dpi

dt= �−

pk

mkpi + Fpi

� , �A1�

from t=0 to t=�. Integrating, we have

−mk

pk�ln�−

pk

mkpi + Fpi

��0

�

= � , �A2�

giving

pi�� exp��−pk

mkpi + Fpi

� �

�pipi, �A3a�

=pie−�pk/mk +

mk

pkFpi

�1 − e−�pk/mk� , �A3b�

=pie−�pk/mk + �Fpi

e−�pk/2mk

sinh�pk

2mk

�pk

2mk

. �A3c�

APPENDIX B: INVARIANT MEASURE OF THE BKPHASE SPACE FLOWS

The phase space compressibility of the phase space BKthermostat is

�BK =�Bij

BK

�xi

�HBK

�xi= −

1

m�

�G1

�p

�K1

�p�

−1

m

�G2

�q

�K2

�p

.

�B1�

Upon introducing the function

HTBK = H +

K1

m�

+K2

m

, �B2�

one can easily find that

�BK =1

kBT

dHTBK

dt, �B3�

so that the invariant measure in phase space reads

d = dx exp− �t

dt�BK , �B4a�

=dx exp�− �HTBK� , �B4b�

=dx exp�− �HBK�exp�� + � , �B4c�

as desired.

APPENDIX C: INVARIANT MEASURE OF THE BKNHPHASE SPACE FLOWS

The phase space compressibility of the NH controlledBulgac-Kusnezov thermostat is

�BKNH =�Bij

BKNH

�xi

�HBKNH

�xi

= −1

m�

�G1

�p

�K1

�p�

−1

m

�G2

�q

�K2

�p

− 2p

m

. �C1�


HTBKNH = H +

K1

m�

+K2

m

+p

2

2m

, �C2�

we have

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.5 4

P

r

-4-3-2-101234

-4-3-2-1 0 1 2 3 4

p

q

FIG. 4. Radial phase space probability for a harmonic oscillatorunder Bulgac-Kusnezov-Nosé-Hoover chain dynamics. The con-tinuous line shows the theoretical value while the black bullets dis-play the numerical results. The inset displays a plot of the phasespace distribution of points along the single trajectory used to com-pute the radial probability.


036705-12

�BKNH =1

kBT

dHTBK

dt, �C3�

so that the invariant measure in phase space is

d = dx exp− �t

dt�BKNH , �C4a�

=dx exp�− �HTBKNH� , �C4b�

=dx exp�− �HBKNH�exp�� + + 2� . �C4c�

APPENDIX D: INVARIANT MEASURE OF THE BKNHCCHAIN PHASE SPACE FLOWS

The phase space compressibility of the Nosé-Hoover-Bulgac-Kusnezov chain is

�BKNHC =�Bij

BKNHC

�xi

�HBKNHC

�xi

= −1

m�

�G1

�p

�K1

�p�

−1

m

�G2

�q

�K2

�p

−p

m

−p�

m�

.

�D1�


HTBKNHC = H +

K1

m�

+K2

m

+p

2

2m

+p�

2

2m�

, �D2�

we have

�BKNHC =1

kBT

dHTBK

dt, �D3�

so that the invariant measure in phase space reads

d = dx exp− �t

dt�BKNHC , �D4a�

=dx exp�− �HTBKNHC� , �D4b�

=dx exp�− �HBKNHC�exp�� + + + �� . �D4c�

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036705-13

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�2007�.


036705-14

bulgac-kusnezov-nosé-hoover thermostats · which molecular dynamics simulation at constant...

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