Journal of Modern Physics, 2011, 2, 231-235
doi:10.4236/jmp.2011.24032 Published Online April 2011 (http://www.SciRP.org/journal/jmp)
Schrödinger-Langevin Equation and Ion Transport at
Nano Scale
Samyadeb Bhattacharya1, Suman Dutta2, Sisir Roy1
1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India
2Department of Mathematics, Charuchandra College, Kolkata
Received November10, 2010; revised January 15, 2011; accepted January 20, 2011
Abstract
Schrödinger-Langevin equation has been constructed for the ion-transport for K-ion channel. The stability of
the solutions of this equation has been discussed under various physical situations. This will shed new light
on the ion transport at nano-scale as well as the possibility of ion trapping and quantum information proc-
essing.
Keywords: Schrödinger-Langevin Equation, Stochastic Mechanics, Lyapunov Stability, Ion Transport at
Nano Scale
1. Introduction
Quantum theory was developed to study the behaviour of
isolated or closed dynamical system. Here, the time evo-
lution of such a closed system is represented by a
one-parameter unitary group in Hilbert space. The notion
of Hamiltonian of the closed system is related with the
infinitesimal generator of the group. However, if we
consider the dynamics of an open quantum system where
the energy exchange with the external world is relevant,
the important issue is whether framework of quantum
theory is adequate for the description of such open sys-
tem. Several attempts [1-5] have been made to study the
dynamics of open quantum system so as to understand
diffusion, dissipation and other non-equilibrium phe-
nomena. The framework of stochastic quantization as
proposed by Nelson [6] and subsequently developed by
other people [7-11] seem to be an attractive procedure to
understand the forces which explicitly depend on veloci-
ties. Here, the dissipative potential term as well as a ran-
dom potential term can be incorporated into Schrödinger
equation which might incorporate the thermal and statis-
tical influence of the environment or external world.
Quantum mechanical version of classical Langevin equa-
tion has been developed known as Schrödinger-Langevin
equation which describes the irreversible behavior of
open quantum system based on Nelson stochastic quan-
tization procedure. Recently, attempts have been made to
study the stability of the solutions of S-L Equations [12].
One of the present authors [13] constructed S-L equation
for ion transport through ion-channels using the frame-
work of Nelson stochastic quantization. The stability of
solutions of this type of S-L equation has been studied in
this paper. This will shed new light on the ion transport
at nano scale. To start with, we shall discuss general me-
thods to solve non-linear Schrödinger equation in section
4.1. Then we shall apply this technique to find the stabil-
ity of the solution of our S-L equation in Section 4.2.
Finally, the possible implications are discussed in Sec-
tion 5.
2. Schrödinger-Langevin Equation and it’s
Construction
At first we will explore the idea and construction of
Schrödinger-Langevin equation. Quantum mechanics for
closed systems is well developed and based on firm
footing. It has described so many sub-atomic phenomena
in isolation with elegant ease. But in practise most of the
systems are open, ie they interact with the environment.
Several attempts have been made to incorporate these
‘interactions’ in the Schrödinger equation by means of
some dissipative components. It has been suggested [9]
that stochastic mechanics as developed by Nelson, can
give us some understanding of how one should treat dis-
sipative forces in terms of quantum dynamics. Here one
considers the forces which explicitly depend on veloci-
ties. The simplest of those is the frictional force linearly

232 S. BHATTACHARYA ET AL.
depending upon velocity. So we are in need for a theory
which corresponds to quantized friction. First we will
dwell upon the construction of Schrödinger equation in
terms of Nelson’s stochastic procedure. In this procedure,
it is postulated that a kind of Brownian motion agitates
all particles of matter. We shall assume that every parti-
cle performs a Markov process of the form
   
d= ,dd
x
tbxttt t   (1)
where is a wiener process with

t 



ts  
independent of

x
r whenever . The diffu- rst
sion coefficient is postulated as =2m 
, where is 
the reduced Planck’s constant. b is the mean forward ve-
locity and is the mean backward velocity.
*
b* =2
bb
u

and *
=2
bb
v are referred to as osmotic velocity and
current velocity respectively.
Studying the kinematics of this type of motion, we
come to the 1st moment kind of equations

=2
u
tm


 vv


u (2)
2
1
=2 V
tm m



 
vvvuu u (3)
where =V
F
is an external force acting on the par-
ticle and m is the mass of each particle. We now want to
change the dependent variable as . The
osmotic velocity can be expressed in terms of a gradient
[6]

=exp iRS  

==R
m


u (4)
where

is the density.
Similarly we set the current velocity as a gradient
=S
m
v (5)
Keeping these assumptions, the Equations (2) and (3) can
be changed into a linear partial differential equation, in
fact the Schrödinger equation.

2
2
i=
2Vt tm


 



 (6)

t 
is an arbitrary constant over time.
To prove this, we can put the expression
in the equation and get,

=exp iRS  




22
22
ii= ii
2
RS RSRS
tt m
Vt 








 
Now taking gradient on each side and separating the real
and imaginary part,we come to the 1st moment Equa-
tions (2) and (3). Since we differentiate the energy equa-
tion over space and come to the force equations, it might
not be unique. That is why the arbitrary function


t 
is taken into the equation. By choosing, for each t, the
arbitrary constant in S appropriately we can arrange for


t  to be zero.
Hence the conventional quantum mechanics can be
formulated in terms of stochastic processes. We can then
use either of the schemes in order to get the information
about quantum dynamics.
We shall now see how can we extend stochastic me-
chanics so as to incorporate velocity dependent forces.
Now let us consider the simplest possible velocity de-
pendent frictional force =
d
F
v

 .
Considering this force Equation (3) is modified into
2
1
=2 V
tm m







vvvvu uu
(7)
Now putting the information =S
m

v in Equation (7)
we get

2
1
=2 VS
tm m






 
vvvuu u
(8)
Now from Equations (2) and (8), by the similar proce-
dure followed previously, we get the Schrödinger equa-
tion

2
2
i=
2VS t tm




 


 (9)
Here *
1
=ln
2i
S







. So Equation (9) becomes

2
2
*
i= ln
22i
Vt
tm





 



  (10)
This is one particular type of Schrödinger-Langevin (S-L)
equation. One can also introduce other terms in the S-L
equation. Here, we will concentrate on this particular S-L
equation.
3. Mathematical Foundation of Stability of
the Solution
We check the stability of the solution by the method of
Lyapunov stability analysis. The solution of a linear equ-
ation (in our case linear in time t) is said to be stable if
there exists a scaler function in the neighbor- 
>0Lx
hood of the origin such that dL0
dt in that region. L is
called the Lyapunov function. From a physicist’s point of
view Lyapunov function can be understood as the energy

S. BHATTACHARYA ET AL.
233
of the system. We know that for stable equilibrium the
potential energy is minimum. So if the total time deriva-
tive of the Lyapunov function is negative, it means that
the the rate of change of energy is negative, i.e. the en-
ergy of the system is tending to it’s minimum.
tive frictional force


0
=
d
F
av where * =ln
2i








v
is the velocity.
The Lyapunov function taken as the energy difference
between present state and the stationary state, is ex-
pressed as
Van & Fülop et al. [12] have taken a non-linear S-L
equation of the form

2
21
,= d
222 s
LV

 
EV








 


v
v (12)
2
20
*
i= ln
22i
a
V
tm


 
 

 


(11)
where

is the probability density and
s
E is the en-
ergy of the stationary state. Here V is the conservative potential contribution and
the non-linear term is the contribution due to the dissipa- The total time derivative of L is found to be

2ln
d=d
d2 4
qs d
LVVEA vFV
t


 


 



v
v
v d

(13)
q
V is the quantum potential. Now it is shown that d
d
L
t derivative of L is calculated. The condition for stability
can be derived by stating the total time derivative of L to
be negative.
is found to be negative providing the second term in the
right hand side is negative. Since this term is the rate of
change of energy due to the frictional force, it can be
taken as negative.
4. Schrödinger-Langevin Equation
Following Van & Fülop et al., we take the expectation
value of the energy difference of the present and the sta-
tionary states as the Lyapunov function. Then we calcu-
late the first variation of L and from that the total time
It has been found that Schrodinger-Langevin equation
can be constructed for the case of ionic diffusion along K
ion channels [13]. The Schrodinger-Langevin equation
describing the ionic diffusion is as follows
22
0
2*
12
**
00
11 22
i = ln
22
expdlnexpdln
2i
tt
a
V
tm i x
tt tt
aa
tt
 




 







  
  
 

 

     
 

 
     






(14)
tial V = 0. Here we will try to find the solution of Equation (14)
and check the stability of it by the method of Lyapunov
stability analysis.
Let the solution be of the form
 
i
,=exp ,
x
t 


xt



. [

(x,t) is real]
4.1. Solution of the S-L Equation
Putting in the Equation (14) we get
We will try to find the solution of (14) following the
method developed by Kostin [11] to find the solution of
nonlinear Schrodinger-Langevin equation for the poten-
2
2=0
x
 
 (15)
2
12
000
11 22
1
=exp dexp d
2
tt
tt tt
aa
at
tmx


 
 
  

 t



 

 


 
 


(16)
From Equation (15) we get






01
,=
x
ttx  t (17)
From Equation (16) we get
 
212
00 00
11 22
1
=,exp dexp
2
tt
tt tt
aa
ta xttt
tm
 
 
d

 
  



 


 



 



(18)

S. BHATTACHARYA ET AL.
234
For weak non-markovian process, [] ne-
glecting the terms in the third bracket
12 0
,aa a
 
2
00
,1
=,
2
xt axt
tm
 



0
at
(19)
From Equation (19) we get
  
00
=0expt   (20.1)






  
11 0
2
0
00
0
=0exp
exp
exp d
2
t
tat
as
ats s
m




 (20.2)
When the time constants 12 are considerable in
comparison with 0, ie for strong non-markovian proc-
ess we cannot neglect the terms in the third bracket. In
that case, we find that the solution will be of the form
(, )aa
a
    
0000 0
0
=0expexpexpd
t
tatatas   Ass (21.1)
 
 



 

2
0
110000 0
00
exp
=0expexpdexpexpd
2
tt
as
tatatssatas
m
    

Bss
(21.2)
where A(s) and B(s) are defined as:
  
12
0
00
11 22
=exp dexp d
ss
st st
aa
0
A
stt 
 
   tt




 
 


(22.1)
  
12
1
00
11 22
=exp dexp d
ss
st st
aa
Bstt tt 
 

 
1




 
 



(22.2)
Since

,
x
t  has linear spacial dependence, the ve-
locity

*
i1
=ln=
2
v
mx mx








,
xt and hence
the frictional force is independent of x. So we may con-
clude that frictional force is uniform all over the region
and the flow is a steady one.
4.2. Stability Analysis of S-L Equation
In one dimension the S-L equation can be written as
22
0
2
i= ln
22i
a
tm
x*



 



 


(23)
The stability condition of which is discussed in section
III.
Following the above procedure of stability analysis,
we find that the stability condition holds for our S-L eq-
uation if the following inequality is satisfied.
1
0
01 1
2
0
22
1
dexp
expdd0
t
d
t
d
tt
a
vF xvF xt
a
tt
avFxt





 
 



 
  


 
 


dd
d




(24)
where

0
=
d
F
av is the frictional force.
Now the first term is always negative. For weak
non-Markovian limit 12 0 , the term in the third
bracket is very small compared to the first term. In this
case the above inequality holds. So the equation has sta-
ble solution [12].
(, )aa a
But for strong non-Markovian cases, there are two
possibilities.
1) If the term in the third bracket is itself negative,
then the inequality strictly holds.
1
0
11
2
0
22
expdd
expdd
t
d
t
d
tt
avFxt
tt
avFxt





  




 




 
(25)
2) If the term in the bracket is positive, then also the
inequality can hold if the magnitude of the first term is
greater than that of the term in the bracket.
1
0
01 1
2
0
22
1
d expdd
expdd
t
dd
t
d
tt
a
vF xvF xt
a
tt
avFxt





 
 



 
  





 


(26)
From Equation (25) we get the condition for stability as:


22
2
0 12
21 2
1
0
exp d
exp d
t
t
s
vs
a
a
s
vs









 



(27)
For the case of 1 =2

 , the condition for stability is
1
aa2
1
(28)
Generally, let 2 =n

 . Where n is a positive real num-

S. BHATTACHARYA ET AL.
235
ber. Then


2
1
20 1
22
1
0
exp d
exp d
t
n
t
s
vs
an
a
s
vs









(29)
Now previously we have shown that

*
i1
=ln=
2
vx
mx mx


 




,t
. Again from the
expression of

we get

2
0
=exp2vM aS

where
M is some constant over time. Then we get from the in-
equality
1
2
2
1
a
an


 (30)

is taken to be smaller than 1. where

0
1
12
= expat
n







5. Conclusive Remarks
The above stability analysis of our S-L equation for ion
transport at nano-scale clearly indicates the following:
 In the weak Non-Markovian limit i.e. 012 ,,
the S-L equation has a stable solution and the
process is a reversible Markov process for certain
duration of time 0
aaa

. This clearly shows that Nel-
son process is realizable in nature atleast in bio-
logical domain. This happens in the selectivity fil-
ter of K-ion channel.
 In strong Markovian limit where 12 &aa can not
be neglected, we can get also stable solutions under
the condition given by Equation (30).
It is found from the recent experiments [14] that the
ionic oscillations are found in the selectivity filter of
K
 ion channel for a certain duration of time. Real sit-
uation demands [15] a stable solution of S-L equation for
a certain period. Our analysis confirms the stability of
solutions in the weak non-Markovian limit. In the basket
region of this channel it clearly shows diffusion of
K

ions. It means that the S-L equation should also have a
stable solution in the strong Markovian limit under a
simple proportionality condition 21 =n

 . This kind of
analysis will be helpful in understanding the behaviours
of nano pores and it’s applications in biological domain.
Our proposal of building S-L equation for ion transport
will shed new light not only in the context of nano-scale
engineering but also in ion-trapping which is a challeng-
ing issue for quantum entanglement and build up quan-
tum computer.
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