Alessandro Sergi and Gregory S. Ezra- Bulgac-Kusnezov-Nosé-Hoover thermostats

8/3/2019 Alessandro Sergi and Gregory S. Ezra- Bulgac-Kusnezov-Nos-Hoover thermostats

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Bulgac-Kusnezov-Nos-Hoover thermostats

Alessandro Serg i*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, USAReceived 3 February 2010; published 19 March 2010

In this paper, we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly dened, one where theBulgac-Kusnezov demons are globally controlled by means of a single additional Nos variable, and anotherwhere each demon is coupled to an independent Nos-Hoover thermostat. Numerically stable and efcientmeasure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaoticproperties of the different phase space ows are numerically illustrated through the paradigmatic example of the one-dimensional harmonic oscillator. It is found that, while the simple Bulgac-Kusnezov thermostat isapparently not ergodic, both of the Nos-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. INTRODUCTIONIn condensed matter studies, there are many situations in

which molecular dynamics simulation at constant tempera-ture 13 is needed. For example, this occurs when mag-netic systems are modeled in terms of classical spins 47 .Deterministic methods 810 , based on non-Hamiltoniandynamics 1119 , 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 Nos dynamics cannot beapplied directly. To tackle this problem, Bulgac and Kusn-

ezov BK introduced a deterministic constant-temperaturedynamics 2224 , which can be applied to spins. A numberof numerical approaches to integration of spin dynamics canbe found in the literature 2528 . However, BK dynamics,as any other deterministic canonical phase space ow, 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 difcult to check forstatistical systems with many degrees of freedom, while it isknown that, despite its simplicity, the one-dimensional har-monic oscillator provides a difcult and important challengefor deterministic thermostatting methods 9,2931 .

In this paper, we accomplish two goals. First, by reformu-

lating BK dynamics through non-Hamiltonian brackets14,15 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 efcient measure-preserving and time-reversiblealgorithms in a systematic way for all the phase space owstreated here.

The BK phase space ow introduces temperature controlby means of ctitious 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 Nos-type variables 8 . In one case, the BK demonsare controlled globally by means of a single additional Nos-Hoover thermostat 8,9 . In the following this will be re-ferred to as Bulgac-Kusnezov-Nos-Hoover BKNH dy-namics. In the second case, each demon is coupled to anindependent Nos-Hoover thermostat. This will be called theBulgac-Kusnezov-Nos-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 Noscontrolled BK dynamics is very encouraging for future ap-plications to thermostatted spin systems.

This paper is organized as follows. In Sec. II, we brieysketch the unied formalism for non-Hamiltonian phasespace ows 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 our

conclusions.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 spaceows are derived in Appendices BD, respectively.

II. NON-HAMILTONIAN BRACKETS AND MEASURE-PRESERVING ALGORITHMS

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

*[email protected]@cornell.edu

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Upon introducing an antisymmetric tensor eld general-ized Poisson tensor 21,33 in phase space, B x = B T x ,one can dene non-Hamiltonian brackets 14 16 as

a ,b =i, j=1

2n a xi

B ij b x j

, 1

where a = a x and b = b x are two arbitrary phase spacefunctions. The bracket dened in Eq. 1 is classied as non-Hamiltonian 14 16 since, in general, it does not obey theJacobi relation, i.e., in general the Jacobiator J 0, where21

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 B ij is said to dene an almost-Poisson structure 34such systems have also been called pseudo-Hamiltonian33 .

An energy conserving and in general non-Hamiltonianphase space ow is then dened by the vector eld

xi = xi, H = j=1

2 N

B ij H x j

, 3

where conservation of H x follows directly from the anti-symmetry of B ij .

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 is

invertible this is true for all the cases considered here , withinverse ij , we can dene the 2-form 35

=12 ij

dx i dx j . 4

The dynamics of Eq. 3 is then Hamiltonian if and only if the 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 efcient 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

2 N

B ij H x j

. 6

When the phase space ow has a nonzero compressibility

=i, j=1

2 N B ij xi

H x j

, 7

the statistical mechanics must be formulated in terms of amodied phase space measure 1218

= ew x 8where

= dx1 dx2 . . . dx2 N 9

is the standard phase space volume element volume form35 and the statistical weight w x is dened by

dw

dt = x . 10

It has been shown that, provided the condition

x j

ew x B ij = 0, i = 1, ... 2 N 11

is satised, then

L = 0 for every , 12so that the volume element is invariant under each of the L 19 . The condition 11 is satised 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

ns1

exp

2 L exp exp Ln x

=1

ns1

exp

2 Lns

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

H BK = p2

2m+ V q +

K 1 p m

+K 2 p

m + k BT + 14a

= H q, p + K 1 p m

+ K 2 p m

+ k BT + , 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 2224 . K 1 and K 2 provide the kinetic en-ergy of demon variables, and for the moment are left arbi-trary.

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

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B BK =

0 0 0 1 0 G 2

0 0 0 0 G 1 p

0

0 0 0 0 0 G 2 q

1 0 0 0 G 1 0

0 G 1 p

0 G 1 0 0

G 2 0 G 2 q

0 0 0

, 15

where G 1 and G 2 are functions of system variables p , qonly.

Substituting B BK and H BK into Eq. 3 , we obtain theenergy-conserving equations

q = H p

G 2 q, p

m

K 2 p

, 16a

=1

m

G 1 p

K 1 p

, 16b

=1

m

G 2 q

K 2 p

, 16c

p = H q

G 1 q, p

m

K 1 p

, 16d

p = G1 H p

k BT G 1 p

, 16e

p = G2 H

q k BT

G 2

q. 16f

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

Algorithm for BK dynamicsIn order to derive a measure preserving algorithms, the

rst step, following Eq. 5 , is to introduce a splitting of H BK:

H 1BK = V q , 17a

H 2BK =

p2

2m, 17b

H 3BK = k BT , 17c

H 4BK = k BT , 17d

H 5BK =

K 1 p m

, 17e

H 6BK =

K 2 p m

. 17f

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

L1BK =

V q

p+ G 2

V q

p , 18a

L2BK =

p

m

q+ G 1

p

m

p , 18b

L3BK = k BT

G 1 p

p , 18c

L4BK = k BT

G 2 q

p , 18d

L5BK =

1m

G1 p

K 1 p

G 1m

K 1 p

p, 18e

L6BK =

G 2m

K 2 p

q+

1m

G 2 q

K 2 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 K 1, K 2, G 1, and G 2 must be madein order obtain explicit formulas. In this paper, we make thefollowing simple choices:

G 1 = p , 20a

G 2 = q , 20b

K 1 = p

2

2, 20c

K 2 = p

2

2. 20d

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

B BK =

0 0 0 1 0 q0 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

H BK = H q, p + p 2

2m +

p 2

2m + k BT + . 22

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The split Liouville operators now simplify as follows:

L 1BK = V q

p+ q

V q

p , 23a

L 2BK = p

m

q

+ p2

m

p

, 23b

L 3BK = k BT

p , 23c

L 4BK = k BT

p , 23d

L 5BK = p m

p m

p

p+

p 2

m

p , 23e

L

6BK

= p m q

q + p m

+ p

2

m

p . 23f

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

L ABK L 1BK + L

4BK = F q

p+ F p

p , 24a

L BBK L 2BK + L

3BK =

p

m

q+ F p

p , 24b

LC BK

L

5BK

+ L

6BK

= p m p

p p m q

q + p m

+ p m

,

24c

where

F q = V / q , 25a

F p = q V q

k BT , 25b

F p = p2

m k BT . 25c

Dening

U BK = exp L BK , 26

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

U BK = U BBK

4U C

BK

2U B

BK

4U A

BK

U BBK

4U C

BK 2

U BBK

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 +

4F p

:U BBK

4 ,

ii

p p exp

2

p m

q q exp

2 p m

+

2 p m

+

2

p m

:U C BK

2,

iii

q q +

4

p

m

p p +

4F p

:U BBK

4,

iv p p + F

p p + F p :U A

BK ,

v

q q + 4

pm

+

4

p m

p p +

4F p

:U BBK

4,

vi

p p exp

2

p m

q q exp

2

p

m

+

2 p m

+

2 p m

:U C BK 2,

vii

q q +

4

p

m

p p +

4F p

:U BBK

4.


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IV. BULGAC-KUSNEZOV-NOS-HOOVER DYNAMICS

The BKNH Hamiltonian

H BKNH = H q, p +K 1 p

m +

K 2 p m

+ p

2

2m

+ k BT + + 2k BT 28

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

BKNH

=

0 0 0 0 1 0 G 2 0

0 0 0 0 0 G 1 p

0 0

0 0 0 0 0 0 G 2 q

0

0 0 0 0 0 0 0 1

1 0 0 0 0 G 1 0 0

0 G 1 p

0 0 G 1 0 0 p

G 2 0 G 2 q

0 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

G 2 q, p

m

K 2 p

, 30a

=1

m

G 1 p

K 1 p

, 30b

=1

m

G 2 q

K 2 p

, 30c

= p m

, 30d

p = H

q

G 1 q, p

m

K 1

p , 30e

p = G 1 H p

k BT G1 p

p p m

, 30f

p = G 2 H q

k BT G2 q

p p m

, 30g

p = p m

K 1 p

+ p m

K 2 p

2k BT . 30h

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

Algorithm for BKNH dynamicsThe Hamiltonian can be split as

H 1BKNH = V q , 31a

H 2BKNH =

p2

2m, 31b

H 3BKNH = k BT , 31c

H 4BKNH = k BT , 31d

H 5BKNH =

K 1 p m

, 31e

H 6BKNH =

K 2 p m

, 31f

H 7BKNH

= p

2

2m , 31g

H 8BKNH = 2k BT , 31h

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

L = B ijBKNH

H BKNH

x j

xi, 32

yields

L1BKNH =

V q

p+ G 2

V q

p , 33a

L2BKNH =

p

m

q+ G 1

p

m

p , 33b

L3BKNH = k BT

G 1 p

p , 33c

L4BKNH = k BT

G 2 q

p , 33d

L5BKNH =

1m

G 1 p

K 1 p

G 1m

K 1 p

p+

p m

K 1 p

p ,

33e

L6BKNH =

G 2m

K 2 p

q+

1m

G 2 q

K 2 p

+

p m

K 2 p

p ,

33f

L7BKNH =

p m

p m

p

p

p m

p

p , 33g

L8BKNH = 2 k BT

p . 33h


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At this stage, we leave the general formulation and adoptthe particular choice of K 1, K 2, G 1, and G 2 given in Eq. 20 .The antisymmetric BKNH tensor becomes

B BKNH =

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 simplies to

H BKNH = H q, p + p 2

2m +

p 2

2m +

p 2

2m + k BT + + 2 BT .

35The split Liouville operators are now

L 1BKNH = V q

p+ q

V q

p , 36a

L 2BKNH = p

m

q+

p2

m

p , 36b

L 3BKNH = k BT

p , 36c

L 4BKNH = k BT

p , 36d

L 5BKNH = p m

p m

p

p+

p 2

m

p , 36e

L 6BKNH = p m

q

q+

p m

+

p 2

m

p , 36f

L 7BKNH = p m

p m

p

p

p m

p

p , 36g

L 8BKNH = 2 k BT

p . 36h

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

L ABKNH L 1BKNH + L

4BKNH + L

7BKNH

= F q

p+

p m

p m

p

p +

p m

p + F p

p 37a

L BBKNH L 2BKNH + L

3BKNH =

p

m

q+ F p

p 37b

LC BKNH L 5BKNH + L

6BKNH + L

8BKNH

= p

m p

p

p

m q

q+

p

m

+

p

m

+ F p

p ,

37c

where

F q = V q

, 38a

F p = q V q

k BT , 38b

F p = p2

m k BT , 38c

F p =

p 2

m +

p 2

m 2k BT . 38d

In L A there appears an operator with the form

Li = pk mk

pi + F pi

pi, 39

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

e Li

pi = p ie pk / mk

+ F pie 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-ned as

U BKNH = exp L BKNH , 42where = A , B , C . One possible reversible measure-preserving integration algorithm for the BKNH thermostatcan then be derived from the following Trotter factorization:

U BKNH = U BBKNH

4U C

BKNH

2U B

BKNH

4U A

BKNH

U BBKNH

4U C

BKNH

2U B

BKNH

4. 43

The direct translation technique gives the followingpseudocode:


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i

q q +

4 p

m

p p +

4F p

:U BBKNH

4,

ii

p p exp

2

p

m

q q exp

2

p m

+

2

p m

+

2 p m

p p +

2F p

:U C BKNH

2,

iiiq q +

4 p

m

p p +

4F p

:U BBKNH

4,

iv

p p + F q

p p + F p

+ p m

p p exp p m

:U ABKNH ,

v

q q +

4 p

m

p p +

4F p

:U BBKNH

4,

vi

p p exp

2 p m

q q exp

2

p m

+

2

p

m

+

2 p m

p p +

2F p

:U C BKNH

2,

vii

q q +

4 p

m

p p +

4F p

:U BBKNH

4.

V. BULGAC-KUSNEZOV-NOS-HOOVER CHAIN

For simplicity, we explicitly treat only the case, in whichthe p and p demons are each coupled to a standard Nos-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. Denethe Hamiltonian

H BKNHC = H q, p +K 1 p

m +

K 2 p m

+ p

2

2m +

p 2

2m

+ k BT + + + . 44

Upon dening 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

B BKNHC =

0 0 0 0 0 1 0 G 2 0 0

0 0 0 0 0 0 G 1 p

0 0 0

0 0 0 0 0 0 0 G 2 q

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 G 1 0 0 0

0 G 1 p

0 0 0 G1 0 0 p 0

G 2 0 G 2 q

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

, 45


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associated non-Hamiltonian equations of motion are

xi = B ijBKNHC

H BKNHC

x j46

withi=1 , . . . , 10 .

Algorithm for BKNHC chain dynamics

Splitting the BKNHC chain Hamiltonian as

H 1BKNHC = V q , 47a

H 2BKNHC =

p2

2m, 47b

H 3BKNHC = k BT , 47c

H 4BKNHC = k BT , 47d

H 5BKNHC =

K 1 p m

, 47e

H 6BKNHC =

K 2 p

m , 47f

H 7BKNHC =

p 2

2m , 47g

H 8BKNHC = k BT , 47h

H 9BKNHC =

p 2

2m , 47i

H 10BKNHC = k BT , 47j

we obtain the corresponding measure-preserving splitting of the Liouville operator

L = B ijBKNHC

H BKNHC

x j

xi. 48

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

B BKNHC

=

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

H BKNHC = H q , p + p 2

2m +

p 2

2m +

p 2

2m +

p 2

2m + k BT + + + 50

and associated Liouville operators

L 1BKNHC = V q

p+ q

V q

p , 51a

L 2BKNHC = p

m

q+

p2

m

p , 51b

L 3BKNHC = k BT

p , 51c

L 4BKNHC = k BT p , 51d

L 5BKNHC = p m

p m

p

p+

p 2

m

p , 51e

L 6BKNHC = p m

q

q+

p m

+

p 2

m

p , 51f

L 7BKNHC = p m

p m

p

p , 51g

L 8BKNHC = k BT

p

, 51h

L 9BKNHC = p m

p m

p

p , 51i

L 10BKNHC = k BT

p . 51j

We combine commuting Liouville operators as follows:

L ABKNHC L 1BKNHC + L

4BKNHC + L

9BKNHC

= F q

p+

p m

+

p m

p + F p

p , 52a


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L BBKNHC L 2BKNHC + L

3BKNHC + L 7BKNHC

= p

m

q+

p m

+

p m

p + F p

p , 52b

LC BKNHC L 5BKNHC + L

6BKNHC + L 8BKNHC + L

10BKNHC

= p m

p

p

p m

q

q+

p m

+ p m

+ F p

p + F p

p , 52c

whereF q =

V q

, 53a

F p = q V q

k BT , 53b

F p =

p2

m k BT , 53c

F p =

p 2

m k BT , 53d

F p = p

2

m k BT . 53e

Both in L ABKNHC and L B

BKNHC there appears an operator withthe form

Li = pk mk

pi + F pi

pi, 54

where k , i = , for L A and k , i = , for L B. Againfollowing the derivation in Appendix A, we nd

e Li pi= p

ie pk / mk + F

pie 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

= U BBKNHC

4 U C BKNHC

2 U BBKNHC

4

U ABKNHC

U BBKNHC

4U C

BKNHC

2U B

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 +

4F p e

/ 4 p / 2m

4

p 2m

1

sinh

4

p 2m

:U BBKNHC

4,

ii

p p exp

2

p m

q q exp

2 p m

+

2 p m

+

2 p m

p p +

2F p

p p +

2F p

:U C BKNHC

2,


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iii

q q +

4 p

m

+

4

p m

p p e / 4 p / m +

4F p e

/ 4 p / 2m

4

p

2m

1

sinh

4

p

2m

:U BBKNHC

4,

iv

p p + F

+ p m

p p e p / m + F p e

p / 2m p 2m

1

sinh p 2m

:U ABKNHC ,

v

q q +

4

p

m

+

4

p

m

p p e / 4 p / m +

4F p e

/ 4 p / 2m

4 p 2m

1

sinh

4 p 2m

:U BBKNHC

4,

vi

p p exp

2 p m

q q exp

2

p m

+

2

p m

+

2 p m

p p +

2F p

p p +

2F p

:U C BKNHC

2,

vii

q q +

4

p

m

+

4

p m

p p e / 4 p / m +

4F p e

/ 4 p / 2m 4

p 2m

1

sinh 4

p 2m

:U BBKNHC

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 spaceows and the correct sampling of the canonical distribution.It is well known that it is necessary to generalize basic Nos-

Hoover dynamics 1,8,9 to thermostats such as the Nos-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


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23,24 . To ensure efcient 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 of the 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 justied 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 models38 . 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 k BT were taken to be unity.The time step in all cases was =0.0025, and all runs werecalculated for 10 6 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 ow is not capable of producing 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 produce

canonical 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 ows, 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 Nos-type variable in theBKNH thermostat effectively introduces additional couplingbetween the two demon variables. The effectiveness of theBKNH thermostat is consistent with our ndings results notdiscussed here that introduction of explicit coupling be-tween demons in BK thermostat dynamics also leads to ef-cient thermostatting of the harmonic oscillator.

VII. CONCLUSIONS

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

1

1.01

1.02

1.03

1.04

0 500 1000 1 500 2 000 2 500

H

t

HBK

H BKNH

H BKNHC

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 H BK ,NH controlled Bulgac-Kusnezov dynamics H BKNH , and Bulgac-Kusnezov-Nos-Hoover chain dynamics H BKNHC . 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 of points 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 Nos-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.


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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 Nos-Hoover thermostat or independent Nos-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 ow could be carried outby extending the present approach to multimode systemse.g., a chain of oscillators coupled to BK-type demons with

associated NH thermostats corresponding to two differenttemperatures 4143 .

The techniques presented here for derivation and imple-mentation of thermostats have been shown to be efcient 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 equivalent

to solving the evolution equation recall i k

dp idt

= pk mk

pi + F pi , A1

from t =0 to t = . Integrating, we have

mk pk

ln pk mk

pi + F pi0

= , A2

giving

pi exp pk mk

pi + F pi

pi pi , A3a

= pie pk / mk +

mk pk

F pi 1 e pk / mk , A3b

= pie pk / mk + F pie

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 = B ij

BK

xi

H BK

xi=

1

m

G 1

p

K 1

p

1

m

G 2

q

K 2

p .

B1

Upon introducing the function

H TBK = H +

K 1m

+K 2m

, B2

one can easily nd that

BK =1

k BT

dH TBK

dt , B3

so that the invariant measure in phase space reads

d = dx exp t dt BK , B4a

= dx exp H TBK , B4b

= dx exp H BK exp + , B4c

as desired.

APPENDIX C: INVARIANT MEASURE OF THE BKNHPHASE SPACE FLOWS

The phase space compressibility of the NH controlled

Bulgac-Kusnezov thermostat is

BKNH = B ij

BKNH

xi

H BKNH xi

= 1

m

G1 p

K 1 p

1

m

G2 q

K 2 p

2 p m

. C1


H TBKNH = H +

K 1m

+K 2m

+ 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-Nos-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.


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BKNH =1

k BT

dH TBK

dt , C3

so that the invariant measure in phase space is

d = dx exp t dt BKNH , C4a

= dx exp H TBKNH , C4b

= dx exp H BKNH exp + + 2 . C4c

APPENDIX D: INVARIANT MEASURE OF THE BKNHCCHAIN PHASE SPACE FLOWS

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

BKNHC = B ij

BKNHC

xi

H BKNHC xi

= 1

m

G 1 p

K 1 p

1

m

G 2 q

K 2 p

p m

p m

.

D1


H TBKNHC = H +

K 1m

+K 2m

+ p

2

2m +

p 2

2m , D2

we have

BKNHC =1

k BT

dH TBK

dt , D3

so that the invariant measure in phase space reads

d = dx exp t dt BKNHC , D4a

= dx exp H TBKNHC , D4b

= dx exp H BKNHC exp + + + . D4c

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