1609 01603 PDF
1609 01603 PDF
1609 01603 PDF
1
Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
2
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400085, India
3
Savitribai Phule Pune University, Ganeshkhind, Pune-411007, India
Over the last century Bohr van Leuween theorem attracted the notice of physicists. The theorem states
about the absence of magnetization in classical systems in thermal equilibrium. In this paper, we discuss about
fluctuations of magnetic moment in classical systems. In recent years this topic has been investigated intensively
and it is not free from controversy. We a have considered a system consisting of a single particle moving
in a plane. A magnetic field is applied perpendicular to the plane. The system is in contact with a thermal
bath. We have considered three cases: (a) particle moving in a homogeneous medium, (b) particle moving in a
arXiv:1609.01603v1 [cond-mat.stat-mech] 6 Sep 2016
medium with space dependent friction and (c) particle moving in a medium with space dependent temperature.
For all the three cases average magnetic moment and fluctuations in magnetic moment has been calculated.
Average magnetic moment saturates to a finite value in case of free particle but goes to zero when the particle
is confined by a 2-D harmonic potential. Fluctuations in magnetic moment shows universal features in the
presence of arbitrary friction inhomogeneity. For this case the system reaches equilibrium asymptotically. In
case of space dependent temperature profile, the stationary distribution is non-Gibbsian and fluctuations deviate
from universal value for the bounded system only.
2
conceptually. It requires to address the basic problem IV. RESULTS AND DISCUSSIONS
of relative stability of states in nonequilibrium systems
which has been a subject of debate for over several A. Unbound system
decades. The theoretical treatments adopted so far are
mostly phenomenological in nature. Landauer, in a
In Fig.1, we have plotted average magnetic moment
series of papers[20–24], argues that for systems with
hM i as a function of time for unbounded system for
nonuniform temperature the relative stability of two
three forms of space dependent friction at constant
states will be affected by the detailed kinetics all along
temperature. The functional forms of space depen-
the pathways (on the potential surface) between the
dency of friction are: 1. γ(x) = γ0 (1 − λ cos(x/γ1 ))
two states under comparison. It is the effect of thermal
with γ0 = 0.5, λ = 0.9, γ1 = 0.25, 2. γ(x) = γ0 +
fluctuations that plays a crucial role and the resulting
γ1 tanh[(x − γ2 )/γ3 ] with γ0 = 0.5, γ1 = 0.3, γ2 =
effective potential surface may have completely differ-
0, γ3 = 0.1 and 3. γ(x) = γ0 + γ1 tanh[(x − γ2 )/γ3 ]
ent nature from that with uniform temperature. With
with γ0 = 0.5, γ1 = 0.3, γ2 = 0.7, γ3 = 0.1. We no-
the help of his “blowtorch” theorem Landauer shows
tice that after some initial transients, which critically
that a change of temperature away from uniformity
depends on the nature of functional form of friction
even at very unlikely positions of the system on the po-
coefficient, average magnetization asymptotically sat-
tential surface may cause probability currents to set in
urates to a constant value 0.25. It is the same value
moving the system towards a new steady state situation
given by Eq.4.
changing thereby the relative stability of the otherwise
locally stable states.
0.4
Space-dependent friction(cosine)
The variation of friction coefficient in space changes Space-dependent friction(symmetric tanh)
0.35 Space-dependent friction(asymmetric tanh)
the dynamics of the particle in the a potential field but 0.3
Homogeneous medium
0.3
In this section, we focus on numerical results Space-dependent friction (cosine)
Space-dependent friction (symmetric tanh)
obtained by evolving the system using discretised 0.25 Space-dependent friction (asymmetric tanh)
Langevin dynamics with time step dt= 0.001 in the un-
derdamped regime. The medium in which the parti- 0.2
cle is moving is considered to be inhomogeneous. In-
homogeneity arises in two different ways: 1. either P(M) 0.15
friction coefficient (γ) is space dependent or 2. tem-
perature is space dependent. We considered three dif- 0.1
ferent types of space dependency both for friction and
temperature : (A) cosine, (B) symmetric tanh and (C) 0.05
3
In case of space dependent temperature, we have Variable Space dependency hM i Theoretical value
kept the friction coefficient fixed to unity and consid- Cosine 0.27176
Friction
ered same three functional form of space dependency Symmetric tanh 0.2671
(γ)
Asymmetric tanh 0.2758
as that of friction. Here we see from Fig.3 that aver-
Cosine 0.4141 0.25
age magnetic moment saturates at large time to a value Temperature
Symmetric tanh 0.4502
which is different from that given by Eq.4. This is due (T )
Asymmetric tanh 0.4331
to the fact that space dependent temperature drives the Constant friction and temperature 0.2501
system out of equilibrium. Fig.4 depicts the fluctua-
tions of magnetic moment about the saturation value TABLE I. For harmonically unbound system.
which clearly shows the universal behavior.
0.45
Space-dependent temperature(cosine)
0.4 Space-dependent temperature(symmetric tanh)
Space-dependent temperature(asymmetric tanh)
Homogeneous medium
0.35
0.3
0.25
hMi 0.2 B. Bounded system
0.15
0.1
hMi 0.04
0.3
0.02
P(M) 0
0.2
-0.02
0.001 0.01 0.1 1 10 100
0.1
t
0
-30 -20 -10 0 10 20 30 FIG. 5. (Color online) Average magnetic moment for
M bounded system.
4
3 A clear deviation in probability distribution P (M ) in
Equilibrium result
Space-dependent friction (cosine) case of space dependent temperature from that of the
2.5 Space-dependent friction(symmetric tanh) equilibrium system is also seen. Table II summarizes
Space-dependent friction(asymmetric tanh) the results for bounded systems.
2
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