Construction and Test of Thermostats and Twirlers
for Molecular Rotations
Siegfried Hess
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin,
PN 7-1, Hardenbergstr. 36, D-10623 Berlin, Germany
Z. Naturforsch. 58a, 377 –391 (2003); received May 28, 2003
The equations of motion are coupled with a dynamical variable, referred to as twirler, which ran-
domizes the angular momentum. The equations are time-reversal invariant, just as those for the stan-
dard Gaussian, Nos´e-Hoover and configurational thermostats. The derivation of the basic equations
is outlined. Test calculations are performed for the two-dimensional isotropic harmonic oscillator and
for a nonlinear elastic dumbbell, used as a simple model to study properties of polymer molecules.
Graphs of characteristic quantities and orbits, some of which are rather intriguing, are displayed. As
applications, the rotational diffusion and the influence of a shear flow on the angular velocity and the
deformation of the model polymer are analyzed.
Key words: Molecular Rotations; Dumbbell; Thermostats; Diffusion; Shear Flow.
1. Introduction
Time-reversible thermostats such as the Gaussian
isokinetic thermostat and those due to Nos´e and
Hoover [1–5] which imply a behavior consistent with
a canonical distribution in equilibrium, as well as gen-
eralisationsthereof[6,7],includingthe configurational
thermostat [8], have found widespread applications in
equilibrium and non-equilibrium molecular dynamics
computer simulations [1–5,9] for systems containing
many (typically 102to 106) particles. When applied to
systems offew particles in two or threedimensionsit is
desireabletohavea “stirring”,“spinning”or “twirling”
(rotational rocking) mechanism which affects the di-
rections of the velocities and not just their magnitudes
as all standardthermostatsdo.This pointis particularly
evidentforthe extendabledumbbellsubjected to a sim-
ple shear flow which has recently been used as a model
[10] to study the shear induced rotation and deforma-
tion of polymer molecules [11]. A term is needed in
the equation of motion analogous to the Lorentz force
involving a fluctuating magnetic field which obeys an
additional differential equation.
In this article, firstly some generalremarks are made
on the derivation of equations governing thermostats
and on the construction of a dynamic stirring mecha-
nism referred to as “twirler”. Then a special choice for
a twirler or “directional thermostat” is introduced and
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some applications for a two-dimensional isotropic har-
monic oscillator and for the dumbbel problem are pre-
sented. More specifically, it is tested how the desired
properties of the twirler, like randomization of the an-
gular momentum and values of time averages expected
for a canonical distribution, depend on the size of the
coupling coefficient in the additional dynamic equa-
tion. Combinations of the twirler with Gaussian, Nos´e-
Hoover, force-momentum and configurational ther-
mostats are considered. Examples of orbits are dis-
played. These are particularly intriguing for the adia-
batic twirler applied to the harmonic oscillator, some
graphs are presented in the appendix. For the dumb-
bell, it is demonstrated that the twirler, combined with
the Gaussian thermostat, leads to a rotational diffusive
behavior when an average over the initial conditions is
performed. Furthermore, it is shown that it is possible
to treat the influence of a shear flow on the rotational
behavior and on the deformation of a model polymer
molecule by the present method.
The approach introduced here is complementary to
the kinetic theory methods used to study the coupling
between rotational and translational motions in gases
[12–15] liquids [15] and colloidal dispersions [16]. In
Kinetic Theory, one starts from equations which have
theirreversibilityofthe processesstudiedalreadybuild
in,whereashere,as emphasisedbefore,time-reversible
equations are employed. Similarly, a Langevin equa-
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378 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
tion with fluctacting forces would also randomize the
angular momentum. There, however, the irreversibility
of the equation is build in due to the use of a friction
coefficient.
2. On the Construction of Thermostats and
“Twirlers”
2.1. General Remarks
For simplicity one particle in Dspatial dimensions,
with D=2orD=3, is considered. The position and
momentum variables are rand p. The force F=F(r)
acting on the particle is assumed to be derived from an
interaction potential
Φ
which provides characteristic
length and energy scales. With the help of them and
the mass mof the particle, all physical quantities of
interest, including the time tand the temperature T,
can be given as multiples of reference values. For the
dimensionless reduced variables the same symbols are
used as for the correspondingphysical variables. Then,
the equation of motion for the constraint free situation
is equivalent to ˙r=p,˙p=F. Terms associated with a
flow velocity will be added later. Thermostats and the
twirler impose constraints on the dynamics.
The configurational thermostat is a special case of a
“r-thermostat” which modifies the equation for ˙rviz.:
˙r=p−
φ
(r,p)
κ
r.(1)
Here the attention is focused on modifications of the
momentum equation. To include the action of a “p-
thermostat” and a twirler, the equations of motion are
replaced by
˙p=F−
ϕ
(r,p)
α
p+
ψ
(r,p)W×p.(2)
The first additional term in the momentum equation
which is proportional to pcomprises the Gaussian, the
Nos´e-Hoover and the “p4” thermostats as well as the
“force-momentum” thermostat to be introcuced later
as special cases corresponding to specific choices for
the scalar quantity
ϕ
. Equations for the scalar dynami-
cal variables
κ
and
α
are stated later. The additional
term involving the pseudo-vector Wdoes not affect
the magnidute of the momentum but it provides direc-
tional changes corresponding the desired stirring mo-
tion. The scalar
ψ
is specified later. Before equations
of change for the additional variables
α
and Ware
stated, it is noticed that time reversal invariance of the
equation of motion requires that both
ϕα
and
ψ
W
change sign under time reversal. The dynamic vari-
ables
κ
,
α
and Ware determined by arguments sim-
ilar to those put forward earlier [4,6]. The distribution
function feq(r,p,
κ
,
α
,W), in thermal equilibrium, is
assumed to be proportional to
exp(−
β
H)exp(−
κ
2/2)exp(−
α
2/2)exp(−W·W/2),
with
β
=1/Tand the HamiltonianH=p2/2+
Φ
. The
requirement that the equations of motion conserve the
extended canonical distribution feq implies
∂
∂r·(˙rfeq)+ ∂
∂p·(˙pfeq)+ ∂
∂
κ
(˙
κ
feq)
+∂
∂
α
(˙
α
feq)+ ∂
∂W·(˙
Wfeq)=0.
(3)
This condition, together with (1), (2) and the assump-
tion that ˙
κ
,˙
α
and ˙
Wbe independent of
κ
,
α
and W,
leads to
d
κ
dt=−
βφ
F·r−(D
φ
+r·∂
∂r
φ
),(4)
d
α
dt=
βϕ
p2−(D
ϕ
+p·∂
ϕ
∂p),(5)
dW
dt=p×∂
∂p
ψ
.(6)
The coupled equations (1,2) and (4–6) are invariant
under time reversal. The relations for ˙
κ
and ˙
α
are
equivalentto equations presentedearlier [4,6]. The ex-
pression for ˙
Wis new. The special case
ϕ
=const cor-
responds to the Nos´e-Hoover thermostat. Notice that
ψ
=const, onthe other hand,implies W=constwhich
contradicts the above made assumption that the values
of Wobey a Gaussian distribution with zero mean. A
special choices for
φ
,
ϕ
and
ψ
are presented and dis-
cussed next.
2.2. Nos´
e-Hoover, Force-Momentum,
and Configurational Thermostats
The special choice
ϕ
=
ν
NH =const with the re-
laxation frequency coefficient
ν
NH corresponds to the
Nos´e-Hoover thermostat. For this case, the pertinent
dynamic variable
α
is denoted by
α
NH. For large val-
ues of
ν
NH the dynamics approaches that of the (isoki-
netic) Gaussian thermostat for which one sets
ϕα
=
F·p/p2. This relation guarantees p·˙p=0 such that
p2=DT =const. The “p4-thermostat” corresponds
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 379
to
ϕ
=
ν
p4p2/Twith the relaxation frequency
ν
p4, the
pertinent dynamic variable is denoted by
α
p4.
The choice
ϕ
=cfpF·p/Twith the coupling coef-
ficient cfp =const and the dynamic variable
α
fp is re-
ferred to as “force-momentum”(fp) thermostat.
The configurational thermostat corresponds to
τ
cfr−1
Φ
with the relaxation time coefficient
τ
cf. No-
tice that r−1
Φ
r=−F. The extra term in (1) then is
proportional to the force F. In this case the dynamic
variable is denoted by
κ
cf.
2.3. A Specific Twirler and an Effective
Magnetic Field
The simplest meaningful choice for
ψ
is the scalar
productr·p. This implies that ˙
Wbecomesproportional
to the angular momentum L=r×p. However, the
choice
ψ
∼F·pseems to be more appropriate since it
implies that the twirler affects the dynamics only over
distances where the force acts. More specifically, the
ansatz
ψ
=cstF·p/T(7)
is made with the (dimensionless) stirring coupling co-
efficient cst. Then (6) becomes
dW
dt=−(cst/T)F×p.(8)
The term involving the Win the equation of motion
can be written as
cstBeff ×p
with the effective “magnetic field”
Beff =(F·pW −F·Wp)/T.
The coupling coefficient −cst plays the role of a
charge. The second term proportional to pgives no
contribution in the equation of motion, however it is
needed to have a divergencefree effective B-field. This
effective field is derived from an effective vector po-
tential Aeff =−
Φ
W×pwhere
Φ
is recalled as the in-
teraction potential.Notice that Aeff ·p=0 where,in the
absence of an external flow field, pis just the particle
velocity. Thus the action of the stirring cannot be de-
scribed by a standard Lagrangian or Hamiltionian dy-
namics of a charged particle in a magnetic field which
requires A·˙r=0.
2.4. Equations of Motion
The equations of motion to be applied in the follow-
ing are
˙r=p+
τ
cf
κ
cfF,(9)
and, for the stirrer combined with the Gaussian ther-
mostat
˙p=F−(F·p/p2)p+cst(F·p/T)W×p.(10)
For the Nos´e-Hoover and force-momentum ther-
mostats, one has
˙p=F−
ν
NH
α
NHp−cfp
α
fp(F·p/T)p
+cst(F·p/T)W×p.(11)
The dynamic variable Wis govered by (8). The
quanties
α
.. obey the equations
d
α
NH
dt=
ν
NH(
β
p2−D),
d
α
fp
dt=cfp(F·p/T)(
β
p2−(D+1)).
(12)
The variable
κ
cf associated with the configurational
thermostat obeys the equation
d
κ
cf
dt=
τ
cf(F·F/T−
∆Φ
).(13)
The (instantaneous) “configurational” temperature
Tconf is defined by Tconf =F·F/
∆Φ
[8]. Notice that
one has F·F/T−
∆Φ
=
∆Φ
(Tconf/T−1),i.e.
κ
cf
does not change when the configurational temperature
matches the prescribed temperature T. Incidendally,
the Einstein frequency
ω
Eis related to the Laplacian
∆
applied on the potential by D
ω
2
E=
∆Φ
, with the di-
mension D=2or3.
2.5. Remarks on Coupling Coefficients
Therelaxationfrequenciesandcouplingcoefficients
such as
ν
NH and cst have to be chosen such that physi-
cally meaningful results are obtained, mostly intuition
as well as trial and error have been employed. An ed-
ucated guess for an estimate of the stirring coupling
coefficient cst is made next. Firstly, the case is consid-
ered where the stirrer is combined with the Gaussian
thermostat. Assuming that Fis a central force, the time
change of the angular momentum Lis given by
˙
L=−F·pL/p2+(cst/T)F·pr×(W×p).
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380 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
The long time average ...of both sides of this equa-
tion vanishes, the same holds true for Land W. As-
suming that the r.h.s. of the above equation is approx-
imately zero one obtains L≈(cst/T)p2r×(W×p).
Squaringboth sides of this relation and averagingthem
leads to L2≈c2
stTD2(D−1)r2W2. For the isoki-
netic case analyzed here p2n=(DT)nwas used. Fur-
thermore, the component of Wparallalel to pshould
not be counted, i.e. W·p=0, and one has W2=
D−1. Since L2=
θ
Twith the moment of inertia
θ
=(2/D)r2, these considerations lead to
cst ≈(D2−D)−1D/2.
Thus the recommended value for the stirring coupling
coefficient is 0.5 and √6/12 ≈0.2 for D=2 and
D=3, respectively. Other criteria for the choice of the
coupling coefficient are discussed later.
For the Nos´e-Hoover and the fp-thermostats the
change of the angular momentum reads
˙
L=−(
ν
NH
α
NH +(cfp/T)F·p
α
fp)L
+(cst/T)F·pr×(W×p).
Assuming,as above, ˙
L≈0, squaring andaveragingthe
terms on the r.h.s. of the angular momentum balance
leads to
(
ν
NH
α
NH +(cfp/T)F·p
α
fp)2L2
≈c2
str2F2(D+2)(D−1)2/D.
Isotropy is assumed, e.g. (F·p)2is replaced by
F2p2/D. For the canonical velocity distribution con-
sidered now one has p2n=D(D+2)...(D+n−
1)Tn. Due to
α
2
NH=
α
2
fp=1 and
α
NH
α
fp=0,
and assuming that the averages of the products F2L2
can be factorized, one obtains
ν
2
NH +c2
fpF2/T≈c2
st(r2F2/r2)
·(D+2)(D−1)2/(2T).
In equilibrium, F2/T=
∆Φ
=D
ω
2
Eholds true
where
ω
Eis the Einstein frequency.
The recommended value for cst depends on the
values chosen for
ν
NH and cfp. For the non-
linear elastic dumbbell considered here one has
r2F2≈r2F2=r2
∆Φ
Tand
∆Φ
≈6.
Then, for D=2, the relation given above implies
cst ≈(
ν
2
NH/12+c2
fp/2)1/2. In particular, for
ν
NH =1,
√3,3,10 on has cst =0.289,0.5,0.866,2.887 and
cst =0.764,0.886,1.118,2.972 when cfp is put equal
to 0 and 1, respectively. These and other values of the
couplingcoefficientshavebeen tested,some results are
presented next.
3. Equilibrium Test Calculations
3.1. Model Potentials
Test calculations were performed for the harmonic
oscillator with the potential function
Φ
=r2/2. The
main attention is focussed on a nonlinear dumbbell
model used previously. It intendes to mimic the dy-
namics and the shape of a polymer molecule by that
of a particle at position rwhere r= 0 corresponds to
the center of mass of the polymer molecule. The force
Facting on this particle can be chosen such that the
time average of r2coincides with the mean square ra-
dius of gyration of the polymer coil in equilibrium. In
particular, the force is chosen as
F=−rr−1
Φ
,−
Φ
=r−3−r3,(14)
where
Φ
is the derivative of the potential function
Φ
(r)=(1/2)r−2+(1/4)r4−3/4 with respect to r.
The potential is of nonlinear elastic type with a min-
imum at r=1, in reduced units. The equilibrium ra-
dius of gyration r0is somewhat larger, depending on
the temperature T. In particular one has r0≈1.02 in
two dimensions and for T=0.1, the temperature used
in the previous studies. Here results will be presented
for temperatures ranging from 0.05 to 2.0. The quan-
tity
∆Φ
which sets the scale for a frequency squared
approaches,at low temperatures, the values 2 and 6 for
the harmonic oscillator, in two dimensions, and for the
dumbbell.
3.2. Twirler and Gaussian Thermostat
Some selected results from many test runs, per-
formed with the Runge-Kutta version of NDSolve of
Mathematica, are displayed in the following figures.
In particular, Fig. 1 shows the orbit and the angular
momentum versus time of the harmonic oscillator sub-
jected to the Gaussian thermostat,with the temperature
T=1.0, in reduced units. The upper and lower graph
are for the stirring coupling coefficients cst =0.0 and
0.1, repectively. The orbits are shown over 500 time
units, the angular momentum over the last 180 time
units only. Notice that the angular momentum does not
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 381
Fig. 1. The orbit and the angular momentum
versus time of the harmonic oscillator sub-
jected to the Gaussian thermostat. The tem-
peratureisT=1.0inreducedunits.Theupper
and lower graph are for the stirring coupling
coefficients cst =0.0 and 0.1, repectively. The
orbits are shown over 500 time units, the an-
gular momentum over the last 180 time units
only.
Fig. 2. Sameas the previous figure, but forthe
stirringcouplingcoefficientscst =0.5and1.0.
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382 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
Fig. 3. The orbit and the angular momentum
versus time of the nonlinear elastic dumb-
bellsubjectedtotheGaussianthermostat.The
temperature is T= 0.1 in reduced units. The
upper and lower graph are for the stirring
coupling coefficients cst = 0.0 and 0.2, repec-
tively. The orbits are shown over about 500
time units, the angular momentum over the
last 100 time units only.
Fig. 4. Sameas the previous figure, but forthe
stirringcouplingcoefficientscst =0.5and0.7.
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 383
Fig. 5. The orbits of the harmonic oscillator subjected to the Gaussian thermostat and a twirler. The temperature is T= 1.0
in reduced units. The coupling stirring coefficients are cst = 0.175, 0.2, and 0.225, from left to right. The orbits are shown
over about 300 time units.
Fig. 6. Sameas theprevious figure, but now for thetemperature T= 0.1 and for the stirring coupling coefficients c
st =0.595,
0.6 and 0.605. The orbits are shown over about 600 time units.
change sign for the pure Gaussian thermostat corre-
sponding to cst =0.0. It does change sign when the
twirler is turned on, i.e. for cst =0.0. Analogous re-
sults for the stirring coupling coefficients cst =0.5 and
1.0 are depicted in Figure 2. Corresponding graphs for
the orbit and the angular momentum versus time of
the nonlinear elastic dumbbell subjected to the Gaus-
sian thermostat, with the temperature is T=0.1, in
reduced units, are shown in Figure 3. The upper and
lower graph are for the stirring coupling coefficients
cst =0.0 and 0.2, repectively. The orbits are shown
over about 500 time units, the angular momentum over
the last 100time units only. Similar resultsfor cst =0.5
and0.7 arepresentedin Figure4. Theinitial conditions
chosenwerex=1.0, y=0.0,vx=√T,and vy=−√T,
both for the harmonic oscillator and for the dumbbell.
In general, the twirler induces rather irregular mo-
tions and the angular momentum alternates around
zero for nonzero stirring coupling coefficient, tested in
the range 0.01to 2.0. For certain values of the coupling
coefficient cst, however, periodic orbits are observed,
which look rather intriguing. Examples are shown for
the harmonic oscillator in Fig. 5 and Fig. 6 where the
temperature is 1.0 and 0.1, respectively. In the first
case, there occurs a double-brezel like structure in a
relatively wide interval of couplingcoefficients around
0.2. In the second case, one observes a more complex
orbit in very narrow range of values of cst around 0.6.
Plots of the angular momentum do not easily reveal
characteristic differences between the irregular and the
periodic orbits. Of course, quantities which are sensi-
tive to the “shape” of the orbit like the components 2xy
orx2−y2of the gyrationtensorare noticeablydifferent
for periodic and for irregular orbits.
To test the quality of the twirler, the average of the
angular momentum lzand of its square, more specif-
ically, lzin units of √TGr, and l2
zdivided by its
equilibrium value TGrminus 1, where Gr=r2=
x2+y2is the mean square radius, are plotted in
the upper graph of Fig. 7 as functions of the stir-
ring coupling coefficient cst. The data are extracted
for the last 300 time units of a total of about 350.
The lower graph shows x2−y2and 2xy, divided
by Gr. For a good twirler, all quantities displayed in
Fig. 7 should be close to zero. A good choice for
the coupling coefficient is cst =0.525 where one has
lz/√TGr=−0.0023, l2
z/(TGr)−1=−0.0056,
2xy/Gr=0.00025, x2−y2/Gr=−0.0041. Aver-
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384 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
Fig. 7. The upper graph shows the angular momentum
(black dots) and the square of the angular momentum (gray
circles), the lower one the components of the gyration ten-
sor characterizing its anisotropy, viz. x2−y2(black dots)
and 2xy (gray circles) versus the stirring coupling coeffi-
cient, for the harmonic oscillator subjected to the Gaussian
thermostat (T= 1) and the twirler. For the scaling of these
quantities see the text.
aged over the last 3000 time units of a total of about
3500,onefinds lz/√TGr=−0.00004,l2
z/(TGr)−
1=0.0079, 2xy/Gr=−0.00046, x2−y2/Gr=
−0.00006. Thus these quantities are practically zero,
within computational accuracy, as expected for a har-
monic oscillator in thermal equilibrium. In both cases
the initial conditions were x=√2T,y=0.0, vx=
√T, and vy=−√T. The data were extracted at 1000
or 10000 times seperated by the time interval
δ
t=
0.1
π
/√T, corresponding to
δ
t=0.314159 for T=1.
The square root of the mean square radius, however,
is larger than its equilibrium value by a factor of about
1.4. For the pure Gaussian thermostat (cst =0), this
factor is 1.15. With the help of an additional configura-
tional thermostat the mean square radius can be made
to approach its equilibrium value. Then, however, l2
z
deviates more from its equilibrium value. No system-
atic search for the optimum values of cst and of the
effective relaxation time
τ
cf, cf. (9), was conducted for
the twirler combined with the Gaussian and configura-
tional thermostats.
The dependence of the effect of the twirler on the
correspondingaverages forthe nonlinearelastic dumb-
bell was analyzed in a similar fashion. The initial con-
ditionswere x=1.0,y=0.0,vx=√T, andvy=−√T.
Table 1. Averages of characteristic quantities, evaluated for
thenonlinearelasticdumbbell,presentedforselectedvalues
of the coupling coefficient cst.
cst lz/√TGrl2
z/(TGr)−12xy/Grx2−y2/Gr
0.15 0.0096 −0.023 0.022 0.019
0.17 −0.0060 −0.002 −0.005 0.002
0.25 0.0163 0.074 −0.009 −0.021
0.35 0.0602 −0.004 0.038 0.030
0.48 −0.0201 0.002 0.057 0.003
0.50 −0.0698 −0.047 −0.031 −0.058
0.56 −0.0566 −0.011 −0.037 0.067
Averages wereevalutedover the last 1000time units of
a total of about 1100. More precisely, the data were
extracted at 1000 times separated by the time inter-
val
δ
t=0.1
π
/√T, corresponding to
δ
t=0.993459
for T=0.1. Graphs corresponding to those shown in
Fig. 7 are not displayed for sake of brevity. Good re-
sults are obtained for specific values of the coupling
coefficient cst only, some selected data are listed in
Table 1. Here, a good recommendation for the choice
of the coupling coefficient is cst =0.17. The tempting
value cst =1.0 should not be chosen since there oc-
cur peculiar periodic orbits for coupling coefficients in
the vicinity of 1.0 leading to nonzero averages for the
angular momentum.
The problem with an increased value of Gr=r2
is not so severe for the nonlinear elastic dumbbell due
to the existence of a minimum of the potential at a fi-
nite distance. To be more specific, the square root of
the mean square radius is equal to 1.04 and 1.06 for
the pure Gaussian thermostat with cst =0 and the ad-
ditional twirler with cst =0.17. The equilibrium value
is 1.01.
3.3. Twirler, Nos´
e-Hoover and Force-Momentum
Thermostats
Numerous calculations which test the effectiveness
of the twirler have been performed for the nonlinear
elastic dumbbell subjected to the Nos´e-Hoover and
force-momentum thermostats. The temperature pre-
scribed was T=0.1. For the Nos´e-Hoover relaxation
frequency
ν
NH and the force-momentum coupling co-
efficientcfp,severalvalueswere chosen,andthetwirler
coupling coefficient cst was varied systematically be-
tween 0.01 and 1.0, in some cases up to 3.0, in steps
of 0.01. In some selected intervals, steps of 0.001 were
taken. Averages of the characteristic quantities
lz∗=lz/TGr,l2
z∗=l2
z/(TGr)−1,(15)
2G∗
+=2xy/Gr,G∗
−=x2−y2/Gr,(16)
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 385
ν
NH cfp cst lz∗l2
z∗2G∗
+G∗
−Grp2∗p4∗
10.00.00.12 0.0079 −0.218 0.017 0.030 1.066 0.039 0.005
10.00.00.71 0.0003 0.103 −0.008 0.029 1.059 −0.038 0.015
10.00.02.884∗0.0061 −0.012 −0.093 0.031 1.070 0.012 −0.077
10.01.00.06 −0.0061 −0.100 0.004 0.006 1.098 0.026 0.232
10.01.00.96 0.0015 −0.100 −0.015 −0.067 1.055 −0.043 0.014
10.01.02.97∗−0.0053 0.045 −0.054 0.082 1.062 −0.029 −0.034
3.00.00.09 −0.0011 0.119 −0.003 0.002 1.069 −0.055 −0.211
3.00.00.56 0.0080 0.081 −0.012 0.001 1.062 −0.009 −0.036
3.00.00.86∗−0.0033 −0.038 −0.062 0.119 1.064 −0.083 −0.081
3.01.00.35 0.0004 0.006 −0.076 −0.043 1.061 0.005 −0.046
3.01.00.46 −0.0023 −0.001 −0.032 0.040 1.062 0.009 −0.033
3.01.01.11∗−0.0043 −0.006 0.068 0.106 1.054 0.013 0.097
√30.00.13 −0.0072 −0.053 −0.050 0.007 1.080 −0.001 −0.019
√30.00.24 −0.0051 −0.111 0.054 0.011 1.065 0.001 −0.042
√30.00.50∗0.0069 0.120 −0.032 −0.078 1.052 0.015 0.136
√30.00.78 −0.0003 0.021 −0.062 −0.016 1.083 −0.003 −0.018
√3√2/40.54∗−0.0024 0.159 −0.081 −0.033 1.042 −0.009 −0.064
√31.00.19 0.0087 0.075 0.042 −0.014 1.063 0.015 0.025
√31.00.88∗0.0098 0.013 0.038 0.017 1.058 −0.013 −0.069
1.00.00.86 0.0051 0.024 0.002 −0.031 1.070 0.003 −0.021
1.01.00.26 −0.0001 −0.134 −0.004 0.014 1.072 0.025 −0.051
1.01.00.40 0.0089 0.062 0.007 −0.006 1.073 0.026 −0.065
1.01.00.51 0.0004 −0.004 −0.010 −0.053 1.059 −0.033 0.047
1.01.00.76∗−0.0070 −0.155 −0.082 0.061 1.066 −0.026 −0.165
0.01.00.30 0.0053 −0.035 −0.065 −0.026 1.092 0.03 −0.293
Table2.Averagesofcharacter-
istic quantities, evaluated for
the nonlinear elastic dumbbell
for T= 0.1, presented for se-
lected values of the relaxation
frequency and coupling coeffi-
cients
ν
NH,cfp, and cst.
Fig. 8. The upper graph shows the angular momentum
(black dots) and the square of the angular momentum (gray
circles), the lower one the components of the gyration ten-
sor characterizing its anisotropy, viz. x2−y2(black dots)
and 2xy (gray circles) versus the stirring coupling coeffi-
cient, for the adiabatic harmonic oscillator with the energy
E=2.0, subjected to the twirler. For the scaling of these
quantities see the text.
and
p2∗=p2/(2T)−1,p4∗=p4/(8T2)−1,(17)
were computed over the last 1000 time units of a to-
tal of about 1100. The initial conditions were chosen
and the extraction of the data was done just as in the
caseof the Gaussian thermostatdiscussed above.Some
results for these averages which should be close to
zero in thermal equilibrium are presented in Table 2.
Also listed are values for Gr=x2+y2which should
be close to 1.02. Data are shown for some of those
coupling coefficients cst where the magnitude of the
average scaled angular momentum lz∗is less than
1/100. The values of cst marked by a star ∗are close
to those values recommended by the considerations
given above. These recommendedvalues give satisfac-
tory results, however, there are combinations of relax-
ation frequency and coupling coefficients which seem
to work better.
Notice that the last row ofTable 2 shows data for the
twirler combined with the force-momentum thermo-
stat but with the Nos´e-Hoover thermostat turned off.
Remarkably enough, the average of p2is close to its
equilibrium value for all cfp larger than 0.1, whereas
the average of p4is systematically too small.
A word of caution is in order. The length of Table 2
seems to imply that practically any value of cst can be
used. However, for many cst, one or more of the mag-
nitude of averages of quantities which should be close
to zero are larger than 1/10. In some cases periodic
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386 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
Fig. 9. The upper graph shows the angular momentum
(black dots) and the square of the angular momentum (gray
circles),theloweronethecomponentsofthegyrationtensor
characterizing its anisotropy, viz. x2−y2(black dots) and
2xy (gray circles) versus the stirring coupling coefficient,
for the nonlinear elastic dumbbell with the energy E= 0.15,
subjected to the twirler. For the scaling of these quantities
see the text.
orbits occur. For
ν
NH =1.0 and cfp =0, the angular
momentum does not average to zero at all values of cst
less than 0.8. For all other combinationsof
ν
NH and cfp
studied relatively large values of the average angular
momentum occur for certain values of cst in a rather
irregular fashion. So before the twirler and the ther-
mostats are applied to the study of specific problems
like the influence of additional forces or the effects of
a flow field, the equilibrium behavior has to be tested.
It is also of interest to study the effect of the twirler
for adiabatic systems where no thermostat is applied.
Some results are presented in the appendix and in
Figs. 8 and 9.
4. Rotational Diffusion
The rotational diffusion coefficient can be inferred
from the mean square angular displacement via an
Einstein relation. The angular displacement
∆ϕ
is the
time integralof the angular velocity
ω
(t)=lz(t)/Gr(t)
where Gris the relevant moment of inertia. The time
integration is approximated by evaluating the angular
velocity in time steps seperated by
δ
t, by performing
the appropriate summation and multiplication by
δ
t.
The value
δ
t=1.0 was chosen in the calculations pre-
Fig. 10. The mean square angular displacement, divided
by 2, as function of the time for a dumbbell subjected to
the Gaussian thermostat with T= 0.1 and a twirler with
the coupling coefficient cst = 0.17. The thin and thick lines
have slopes 2 and 1, respectively, the rotational diffusion
coefficient is ¯
Dr= 0.25.
sented next. The square of the angular displacenent is
averaged over different runs with randomlychosen ini-
tial conditions. A time dependent rotational diffusion
coefficent Dr(t)is defined via the relation
(
∆ϕ
)2=2Dr(t)t.(18)
The “true” rotational coefficient ¯
Dris obtained when
Dr(t)reaches a plateau value at large times. For the
time reversal chaotic dynamics analysed here, one has
to test whether such a diffusive behavior exists at all.
It indeed seems to be the case for the examples studied
here. In Fig. 10 the mean square angular displacement,
divided by 2, is displayed as function of the time for
the nonlinear elastic dumbbell subjected to the Gaus-
sian thermostat with T=0.1 and a twirler where the
coupling coefficient is cst =0.17. The thin and thick
lines have slopes 2 and 1, respectively, the rotational
diffusion coefficient, inferred from the average be-
tweent=500 andt=1000, is ¯
Dr=0.25. The average
...is based on 100 different initial conditions with
the initial positions and velocities x=cos(2
π
Rpos),
y=sin(2
π
Rpos),vx=√2Tcos(2
π
Rvel),vy=√2T=
sin(2
π
Rvel), where Rpos and Rvel are random num-
bers between 0 and 1 generated, within the Mathemat-
ica program, by Random[] with SeedRandom [143].
In Figs. 11 and 12 the rotational diffusion coefficient
Dr(t)is shown as function of the time tfor the non-
linear elastic dumbbell subjected to the Gaussian ther-
mostat, now for T=0.05, and a twirler with the cou-
pling coefficients cst =0.17 and cst =0.5, repectively.
The horizontal lines correspond to the averages evalu-
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 387
Fig. 11. The rotational diffusion coefficient Dr(t)as func-
tion of the time tfor a dumbbell subjected to the Gaussian
thermostat with T= 0.05 and a twirler with the coupling
coefficient cst = 0.17. The horizontal line corresponds to
the average over the last 500 time units, viz.: ¯
Dr= 0.11.
Fig. 12. The rotational diffusion coefficient Dr(t)as func-
tion of the time tfor a dumbbell subjected to the Gaussian
thermostat with T=0.05 and a twirler with the coupling
coefficient cst =0.5.The horizontal line corresponds to the
average over the last 500 time units, viz.: ¯
Dr=0.21.
ated over the last 500 time units, viz.: ¯
Dr=0.11 and
¯
Dr=0.21. The plateau value can be approached from
above, as in Fig. 11 or from below, cf. Figure 12.
As a side remark which is of interest for further ap-
plications,it is mentionedthat, forT=0.1,the average
based on just 10 runs with the initial positions and ve-
locities x=1.0, y=0.0, and vx=√2Tcos(2
π
Rvel),
vy=√2Tsin(2
π
Rvel), where the Rvel are random
numbers generated with the seed mentioned above,
yields practically the same rotational diffusion coeffi-
cients ¯
Dr.
5. Shear-Induced Angular Velocity
The influenceof a shear flow on the angular velocity
of a polymer chain molecule has recently been anal-
ysed in non-equilibrium molecular dynamics (NEMD)
computer simulations of polymer solutions [11]. The
basic features are already seen in simple model calcu-
lations for thermostated nonlinear elastic dumbbells.
In [10], Gaussian, Nos´e-Hoover and configurational
thermostats were used. These thermostats alone do not
randomizethe direction ofthe rotational velocity. Thus
in order to detect the influence of the shear on the an-
gular velocity for shear rates that are small compared
with the thermal angular velocity, the difference in the
behaviorof two dumbbellswas analysed,whichstarted
from equal initial positions but with opposite direc-
tions of their angular velocities. Here test results are
presented for a single dumbbell subjected to a shear
flow; the Gaussian isokinetic thermostat and a twirler
with cst =0.5 are applied.
To be more specific, it is assumed that the dumb-
bell feels a pseudo-friction force −
ζ
(˙r−v(r)) where
v(r)) is the flow velocity of the fluid which, for sim-
plicity, is assumed to be not affected by the presence
of the dumbbell, i.e. the flow acts like an “external”
field. The quantity
ζ
is the pseudo-friction coefficient
of the Gaussian thermostat. In the following it is con-
venient to introduce the “momentum” variable
p=dr
dt−v(r),(19)
which is the peculiar velocity.
Next,the special case ofa planeCouette flow is con-
sidered with the velocity given by v(r)=
γ
yex, where
exis a unitvectorin x-directionand
γ
=∂vx/∂y=const
is the shear rate. Furthermore, as before, the motion of
the particle is restricted to the xy-plane. Then, appart
from the new terms involving W=Wz, the equations
of motion correspondto the (two-dimensionalversion)
of the “SLLOD” algorithm used in NEMD simulation
studies of the viscous properties of fluids [3]:
dx
dt=px+
γ
y,dy
dt=py,(20)
dpx
dt=Fx−
ζ
px−(cst/T)F·pWp
y−
γ
py,(21)
dpy
dt=Fy−
ζ
py+(cst/T)F·pWpx.(22)
The expression for the coefficient
ζ
now is
ζ
=(p·F−
γ
pxpy)/p2,(23)
which guarantees that p2=2T=const. The value of
the temperature Tis fixed by the initial condition. The
time average of
ζ
vanishes for
γ
=0. The same applies
to the variableW. Furthermore (6) reduces to
dW
dt=−(cst/T)(Fxpy−Fypx).(24)
The relation between the shear-induced angular ve-
locity and the influence of the shear flow on the shape
of the orbit, as derived in [10,11] is based on the an-
gular momentum balance and the assumption that the
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388 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
time change of the angularapproacheszero when aver-
aged over long times. For the plane Couette geometry
considered here, this leads to
Lz=−
γ
mGyy.(25)
Theangularbracketindicatesthetime average.The an-
gularvelocity ¯
ω
is defined as the ratio of Land ofthe
relevant component of the (time averaged) moment of
inertia tensor, viz. (Gxx +Gyy). Thus the expressionfor
the angular velocity, inferred from the average angular
momentum is
¯
ω
=Lz
(Gxx+Gyy).(26)
The relation (25) becomes
¯
ω
=¯
ω
G,(27)
where the expression ¯
ω
Gis based on the shape of the
orbit, used to model the geometry of the polymer coil,
which is given by
¯
ω
G=−
γ
Gyy
Gxx+Gyy=−
γ
21−Gxx−Gyy
Gxx+Gyy.
(28)
In an undeformed equilibrium state, the coil is spheri-
cal, on average, then one has ¯
ω
G=−
γ
/2. The same
applies for small shear rates. At intermediate and
at high shear rates, the polymer molecule is substa-
nially deformed such that Gxx >Gyy, on average. This
implies that the ratio
Ω
G=¯
ω
G/(−
γ
)becomes sig-
nificantly smaller than 1/2 which is its small shear
rate limiting value. A relation of the form (27) be-
tween the rotational angular velocity ¯
ω
and the quan-
tity ¯
ω
Gassociated with the deformation of the chain
molecule was first proposed by Cerf [18]. Non-equi-
librium molecular dynamics (NEMD) computer sim-
ulations can and have provided [11] a test of this re-
lation. Here, as in [10], a test is conducted for a sim-
ple two-dimensional dumbbell model. In Fig. 13, the
angular velocity, divided by the negative shear rate,
viz. ¯
ω
/(−
γ
)(black dots) and its geometric counterpart
¯
ω
G/(−
γ
)(larger gray dots) are shown as functions of
the shear rate for a dumbbell subjected to the Gaussian
thermostat and a twirler with the coupling coefficient
cst =0.5. The data were collected over 40/
γ
time units
and averaged over 10 runs with random initial condi-
tions chosen as indicated at the end of the preceding
Fig. 13. The angular velocity, divided by the negative shear
rate (black dots) and its geometric counterpart (larger gray
dots) as functions of the shear rate for a dumbbell subjected
to the Gaussian thermostat and a twirler with the coupling
coefficient cst = 0.5. The dashed horizontal line marks the
value 0.5 expected for a solid-like rotation, the inclined
dashed line corresponds to a power law with the exponent
−1.1.
section. The dashed horizontal line marks the value 0.5
expected for a solid-like rotation, the inclined dashed
line corresponds to a power law with the exponent
−1.1. This value is close to that one found in NEMD
simulations of polymer chains [11]. For cst =0.17 a
steeperdecreaseofthe angularvelocity,with the power
law exponent −2.5, is observed for high shear rates.
The systematic study of the effects of the shear flow on
the orbit and on long time averages for different cou-
pling coefficients cst is outside the scope of this article.
6. Concluding Remarks
In this article, it is demonstrated how the time-
reversible equations of motion can be coupled with
an additional dynamical variable which obeys its own
equation of change such that the angular momentum
of a rotating particle is randomized in equilibrium.
This twirler has been combined with the Gaussian, the
Nos´e-Hoover, the force-momentum and the configura-
tional thermostats. Here, test calculations and applica-
tions to rotational diffusion and to the shear-flow in-
duced modificationsof the rotational motionand of the
shape of the orbit were given for the two-dimensional
isotropic harmonic oscillator and, in particular, for a
nonlinear elastic dumbbell. Some results for the effect
the adiabatic twirler are presented in the appendix.
Alternative time-reversible equations can be formu-
lated when a method, introducedin [6] for the random-
ization of a unit vector (“spin”), is applied to the direc-
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 389
cst lz∗l2
z∗2G∗
+G∗
−Gr/2rsh Ekinp4
2.80 0.0004 0.743 0.001 −0.005 1.000 0.007 1.003 0.702
2.85 −0.0002 0.889 −0.003 −0.006 1.002 0.012 1.002 0.681
2.90 −0.0001 0.977 0.003 0.003 1.002 0.021 1.003 0.665
2.92 0.0004 1.006 0.004 0.002 1.000 0.009 1.005 0.659
2.95 0.0001 1.045 0.046 −0.056 1.003 0.009 1.003 0.653
3.00 0.0002 1.098 −0.009 0.001 1.003 0.025 1.004 0.643
3.05 −0.0006 1.135 −0.006 0.005 1.005 0.023 1.003 0.635
3.10 0.0007 1.188 −0.009 0.008 1.004 0.025 1.004 0.628
Table 3. Averages of characteristic
quantities, evaluated for the adiabatic
harmonic oscillator with E=2.0, pre-
sented for selected values of the cou-
pling coefficient cst.
cst lz∗l2
z∗2G∗
+G∗
−Gr/2rsh Ekinp4
2.14 0.0038 1.104 0.017 −0.052 1.071 0.065 0.106 0.581
2.19 −0.0093 1.171 −0.047 0.023 1.078 0.094 0.110 0.559
2.20 0.0010 1.073 0.046 0.026 1.067 0.185 0.105 0.583
2.29 −0.0071 1.035 0.078 0.087 1.060 0.142 0.099 0.608
Table 4. Averages of characteristic
quantities, evaluated for the adia-
batic nonlinear elastic dumbbell with
E= 0.15, presented for selected values
of the coupling coefficient cst.
tion of the velocity. In that approach, which has been
tested for a doubledumbbell ortrumbbell [18],one has
to choose an anisotropic potential which imposes cer-
tain symmetry properties on the system which should
be distinct from those inherent in the physical problem
studied. The present method seems to be conception-
ally simpler, nevertheless, a comparison between both
methods should be performed.
The extension of test calculations and applications
of the kind considered here to 3-dimensional rotations,
as well as the study of trumbbells [19] and more com-
plex molecular models are desireable. A further appli-
cation is the analysis of the effect of confining walls
on the rotation and, in particular, on the rotational dif-
fusion, as observed in NMR-experiments [20].
Acknowledgements
This work has been conducted under the aus-
pices of the Sonderforschungsbereich SFB 448
“Mesoskopisch strukturierte Verbundsysteme” of the
Deutsche Forschungsgemeinschaft (DFG). Financial
support is gratefully acknowledged. Furthermore, I
thank Bill Hoover, Patrick Ilg, and Martin Kr¨oger for
helpful discussions.
Appendix: Adiabatic Twirler
It is also of interest to test the effect of the twirler
for adiabatic systems where no thermostat is applied.
Results for the harmonic oscillator with the initial con-
ditions x=√2, y=0.0, vx=1, and vy=−1, which
imply the inital kinetic energy Ekin =1.0 and the total
energy E=2.0, are shown in Fig. 8 and in Table 3.
Here the scaled variables
lz∗=lz/EkinGr/2,
l2
z∗=2l2
z/(EkinGr),(29)
and
p4=p4/(2p22)(30)
are used, for G∗
+and G∗
−see (16). Furthermore, rsh =
x2+y2is the shift of the “center of mass” of
the orbit with respect to the origin. The angular mo-
mentum averages to zero for cst >0.1, the quantities
G∗
+and G∗
−characterizing the anisotropy of the or-
bit become close to zero for cst >1.0. A good per-
formance of the twirler is found for cst ≈3, cf. Ta-
ble 3. Here the average of the angular momentum and
G∗
+,G∗
−are rather small. The average kinetic energy
and the mean square radius r2=Grare just slightly
largerthan their initial values. For a particle at distance
rfrom the center and with the tangential momentum
ptan, the instantaneous Lyapunovcoefficients are 0, ±i,
(1/2)(cst/T)wrptan ±(cst/T)2w2r2p2
tan −4. The
Lyapunov exponents
λ
are the averages of these quan-
tities. The estimate where the average of the square
root is replaced by the square root of averages, with
w=0, w2=1, p2
tan=T,r2=2T, leads to
λ
=0,±i,±c2
st/2−1. Thus one of the Lyapunovex-
ponentshas a realpart larger thanzero forcst >√2and
the system behaves in a chaotic manner. These consid-
erations, however,providejust a crude estimate and do
not replace the proper calculation of Lyapunov expo-
nents which is outside the scope of the current inves-
tigation. It should be mentioned that some bewilder-
ing (periodic) orbits are observed for special values of
the coupling coefficient cst, cf. Figure 14. Notice that
several of these orbits, at least over the 200 time units
shown, have broken the rotational symmetry of the in-
teraction potential.
Corresponding results for the nonlinear elastic
dumbbell with the initial conditions x=1.0, y=
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390 S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations
Fig. 14. The orbits of the adiabatic harmonic oscillator subjected to twirlers with selected stirring coupling coefficients c
st
ranging from 0.0 to 4.3. The orbits are shown over 64
π
≈200 time units.
0.0, vx=√0.15, and vy=−√0.15, which imply
the inital kinetic energy Ekin =0.15 and the total energy E=0.15, are presented in Fig. 9 and in
Table 4.
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S. Hess ·Construction and Test of Thermostats and Twirlers for Molecular Rotations 391
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