We derive the generalized Fokker-Planck equation for the probability density function of the nanoparticle
magnetic moment driven by Poisson white noise. Our approach is based on the reduced stochastic
Landau-Lifshitz equation in which this noise is included into the effective magnetic field. We take into account
that the magnetic moment under the noise action can change its direction instantaneously and show
that the generalized equation has an integro-differential form.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3537

Bondar, O.O.

Denysov, Stanislav Ivanovych

Reva, V.V.

Денисов, Станислав Иванович

Денисов, Станіслав Іванович

Electronic Sumy State University Institutional Repository

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE NANOMATERIALS: APPLICATIONS AND PROPERTIES Vol. 1 No 4, 04MFPN15(3pp) (2012)   2304-1862/2012/1(4)04MFPN15(3) 04MFPN15-1  2012 Sumy State University Generalized Fokker-Planck Equation for the Nanoparticle Magnetic Moment  Driven by Poisson White Noise  S.I. Denisov, O.O. Bondar, V.V. Reva  Sumy State University, 2, Rimsky-Korsakov Str., 40007 Sumy, Ukraine  (Received 25 June 2012; published online 26 August 2012)  We derive the generalized Fokker-Planck equation for the probability density function of the nanopar-ticle magnetic moment driven by Poisson white noise. Our approach is based on the reduced stochastic Landau-Lifshitz equation in which this noise is included into the effective magnetic field. We take into ac-count that the magnetic moment under the noise action can change its direction instantaneously and show that the generalized equation has an integro-differential form.  Keywords: Stochastic Landau-Lifshitz equation, Poisson white noise, Probability density function, Gener-alized Fokker-Planck equation.    PACS numbers: 75.78. – n, 05.10.Gg, 05.40.Ca   1. INTRODUCTION  The study of the physical properties of nanoparticle materials, especially their magnetic properties, is of great importance. This is substantiated by the fact that the magnetic nanoparticles and their ensembles are widely used in many areas of science, engineering and medicine [1-3]. Moreover, the investigation of the mag-netic properties of nanomaterials can significantly ex-pand the scope of their application, e.g., in magnetic recording, sensors based on the effect of giant magneto-resistance, magnetic refrigerators, bio-medical technol-ogies, etc. Due to continuous internal and external fluctua-tions that are an integral part of real systems, the dy-namics of the nanoparticle magnetic moment is ran-dom. In some cases, it can be described by the stochas-tic Landau-Lifshitz (LL) equation in which the fluctua-tions are modeled by a white noise. A great amount of effort has been devoted to the study of the role of ther-mal fluctuations approximated by Gaussian white noise. In this case, the nanoparticle magnetic moment evolves as a continuous-time Markov process whose conditional probability density satisfies the ordinary Fokker-Planck (FP) equation [4-6]. Within a given ap-proach, some aspects of the effects of magnetic relaxa-tion, switching of magnetization and induced magneti-zation are considered in Refs. [7-12]. However, although the Gaussian white noise is widely applicable (because of the central limit theo-rem), in some cases the use of white noises with other statistics is more preferable. For example, such a situa-tion occurs when the magnetic moment under strong fluctuations changes its direction so fast that this pro-cess can be considered as instantaneous. To describe this behavior, it is reasonable to use the LL equation in which the fluctuations are modeled by Poisson and/or Lévy noises. The dynamics of the magnetic moment driven by these noises is Markovian and, in contrast to the case of Gaussian white noise, is discontinuous. In this paper, we derive the generalized FP equation asso-ciated with the reduced LL equation driven by Poisson white noise defined as a random sequence of delta pulses whose counting process is Poisson. 2. MODEL AND BASIC EQUATIONS  We deal with a simple model of single-domain fer-romagnetic nanoparticles that are characterized by the magnetic moment  of a constant length . It is assumed that the rotational dy-namics of the vector m is described by the following stochastic LL equation:     (2.1)  where γ(>0) is the gyromagnetic ratio, α(>0) is the damping parameter, the cross denotes the vector prod-uct, is the effective magnetic field act-ing on the magnetic moment, W is the magnetic energy of a particle, and  is a random magnetic field. Introducing the dimensionless time  and ener-gy  (  is a characteristic magnetic field), the LL equation (2.1) in spherical coordinates can be written as a system of two equations    (2.2)  Here,  and  are the polar and azimuth-al angles of m, respectively,    (2.3)  ,   , and   with  being the Cartesian coordinates of the random mag-netic field h. We consider the coordinates  to be independent and identically distributed white noises. Therefore, since these noises are multiplicative, the system of equations (2.2) should be defined more precisely. To this end, we interpret Eqs. (2.2) in the Ito sense [13] (see also Refs. [5,6]). In general, to find the statistical prop-erties of the increments  ( ), it is necessary to solve these equations under condition that the statisti-cal characteristics of  are fixed. However, in order to  S.I. DENISOV, O.O. BONDAR, V.V. REVA PROC. NAP 1, 04MFPN15 (2012)   04MFPN15-2 qualitatively characterize the role of non-Gaussian noises in the dynamics of m, we assume that the statis-tical properties of  are prescribed. According to the definitions [see formulas after Eqs. (2.3)], from the con-dition  (the angular brackets denote an aver-age over the realizations of ) one gets  and, if  (  and , then  (p  and . These properties permit us to consider the increments  as independent and identically dis-tributed variables with zero mean and represent them in the form    , (2.4)  where  are two white noises with identical statisti-cal characteristics. It should be emphasized that such interpretation of noises in Eqs. (2.2) differs from the original one and the stochastic dynamics of m is also different. Nevertheless, we exploit this toy model to get more insight into the effects of non-Gaussian noises. In this work we study the case when  (we omit the index p) is Poisson white noise, i.e., a random se-quence of delta pulses defined as   . (2.5)  Here,  are independent zero-mean random variables with the same probability density q(z), δ(τ) is the δ function,  is the Poisson counting process character-ized by the probability  (  is the rate of the process) that  events occur with-in the interval (0, τ], and  are the event times. For this noise the probability density  that the increment  at  equals  is given by [14]   . (2.6)  According to this result, the probability per unit time that  exceeds any given value is, in general, non-zero in the limit . As a consequence, the dynamics of the magnetic moment m described by Eqs. (2.2) is discontinuous. One more feature of these equations is that , while the polar and azimuthal angles  and  must belong to the in-tervals  and , respectively, for all . There-fore, to find the connection between these angles and the increments  and , we use the relation , which yields    (2.7) and   (2.8)  Here, the whole numbers  and  are chosen for given  and  from the conditions  and . These numbers count the number of rotations of m in the corresponding planes. Thus, Eqs. (2.2) driven by Poisson white noise are now com-pletely defined and we are ready to derive the corre-sponding generalized FP equation. 3. GENERALIZED FOKKER-PLANCK EQUA-TION   The probability density  that  and  is defined in the standard way   . (3.1)  Denoting , from the definition (3.1) and conditions (2.7) and (2.8) one obtains    (3.2)  The average in Eq. (3.2) is equivalent to the average over the variables  and . Taking into account that these two groups of variables are inde-pendent of each other and using Eqs. (2.2), we find            (3.3)       ,   where . Finally, using Eq. (2.6) and the standard definition , after straightforward calculations we arrive at the fol-lowing generalized FP equation:       +     +  (3.4)  +     .   The solution of this integro-differential equation must be normalized, i.e., , and satis-fy the initial condition . Another form of this equation, which can be useful for studying its properties, follows directly from Eq. (3.4) by using the Poisson summation formula.  4. CONCLUSIONS  We have introduced a toy model for studying the ef-fects of Poisson white noise in the rotational dynamics of the nanoparticle magnetic moment. We have shown that the dynamics is discontinuous and have developed an approach to account for the manifold rotations of the magnetic moment under pulses of the noise. Finally, using this approach, we have derived the corresponding generalized Fokker-Planck equation.  ACKNOWLEGMENTS  The authors are grateful to the Ministry of Educa-tion and Science of Ukraine for the financial support.  GENERALIZED FOKKER-PLANCK EQUATION FOR THE NANOPARTICLE… PROC. NAP 1, 04MFPN15 (2012)    04MFPN15-3  REFERENCES  1. J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98, 283 (1997). 2. C.A. Ross, Annu.Rev. Mater. Res. 31, 203 (2001). 3. Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, J. Phys. D: Appl. Phys. 36, R167 (2003). 4. W.F. Brown Jr., Phys. Rev. 130, 1677 (1963). 5. H. Risken, The Fokker-Planck Equation: Methods of Solu-tion and Applications (Berlin: Springer-Verlag 1989). 6. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation (Singapore, WorldScientific, 2004). 7. S.I. Denisov, K.N. Trohidou, Phys. Rev. B 64, 184433 (2001). 8. S.I. Denisov, T.V. Lyutyy, K.N. Trohidou, Phys. Rev. B 67, 014411 (2003). 9. S.I. Denisov, T.V. Lyutyy, P. Hänggi, K.N. Trohidou, Phys. Rev. B 74, 104406 (2006). 10. S.I. Denisov, T.V. Lyutyy, P. Hänggi, Phys. Rev. Lett. 97, 227202 (2006). 11. S.I. Denisov, K. Sakmann, P. Talkner, P. Hänggi, Eu-rophys. Lett. 76, 1001 (2006). 12. S.I. Denisov, A.Yu. Polyakov, T.V. Lyutyy, Phys. Rev. B 84, 174410 (2011). 13. K. Ito, Nagoya Math. J. 1, 35 (1950) 14. S.I. Denisov, W. Horsthemke, P. Hänggi, Eur. Phys. J. B 68, 567 (2009).  

Generalized Fokker-Planck Equation for the Nanoparticle Magnetic Moment Driven by Poisson White Noise

SocioEconomic Challenges Journal (Sumy State University)

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE NANOMATERIALS: APPLICATIONS AND PROPERTIES 
Vol. 1 No 4, 04MFPN15(3pp) (2012) 
 
 
2304-1862/2012/1(4)04MFPN15(3) 04MFPN15-1  2012 Sumy State University 
Generalized Fokker-Planck Equation for the Nanoparticle Magnetic Moment  
Driven by Poisson White Noise 
 
S.I. Denisov, O.O. Bondar, V.V. Reva 
 
Sumy State University, 2, Rimsky-Korsakov Str., 40007 Sumy, Ukraine 
 
(Received 25 June 2012; published online 26 August 2012) 
 
We derive the generalized Fokker-Planck equation for the probability density function of the nanopar-
ticle magnetic moment driven by Poisson white noise. Our approach is based on the reduced stochastic 
Landau-Lifshitz equation in which this noise is included into the effective magnetic field. We take into ac-
count that the magnetic moment under the noise action can change its direction instantaneously and show 
that the generalized equation has an integro-differential form. 
 
Keywords: Stochastic Landau-Lifshitz equation, Poisson white noise, Probability density function, Gener-
alized Fokker-Planck equation. 
 
  PACS numbers: 75.78. – n, 05.10.Gg, 05.40.Ca 
 
 
1. INTRODUCTION 
 
The study of the physical properties of nanoparticle 
materials, especially their magnetic properties, is of 
great importance. This is substantiated by the fact that 
the magnetic nanoparticles and their ensembles are 
widely used in many areas of science, engineering and 
medicine [1-3]. Moreover, the investigation of the mag-
netic properties of nanomaterials can significantly ex-
pand the scope of their application, e.g., in magnetic 
recording, sensors based on the effect of giant magneto-
resistance, magnetic refrigerators, bio-medical technol-
ogies, etc. 
Due to continuous internal and external fluctua-
tions that are an integral part of real systems, the dy-
namics of the nanoparticle magnetic moment is ran-
dom. In some cases, it can be described by the stochas-
tic Landau-Lifshitz (LL) equation in which the fluctua-
tions are modeled by a white noise. A great amount of 
effort has been devoted to the study of the role of ther-
mal fluctuations approximated by Gaussian white 
noise. In this case, the nanoparticle magnetic moment 
evolves as a continuous-time Markov process whose 
conditional probability density satisfies the ordinary 
Fokker-Planck (FP) equation [4-6]. Within a given ap-
proach, some aspects of the effects of magnetic relaxa-
tion, switching of magnetization and induced magneti-
zation are considered in Refs. [7-12]. 
However, although the Gaussian white noise is 
widely applicable (because of the central limit theo-
rem), in some cases the use of white noises with other 
statistics is more preferable. For example, such a situa-
tion occurs when the magnetic moment under strong 
fluctuations changes its direction so fast that this pro-
cess can be considered as instantaneous. To describe 
this behavior, it is reasonable to use the LL equation in 
which the fluctuations are modeled by Poisson and/or 
Lévy noises. The dynamics of the magnetic moment 
driven by these noises is Markovian and, in contrast to 
the case of Gaussian white noise, is discontinuous. In 
this paper, we derive the generalized FP equation asso-
ciated with the reduced LL equation driven by Poisson 
white noise defined as a random sequence of delta 
pulses whose counting process is Poisson. 
2. MODEL AND BASIC EQUATIONS 
 
We deal with a simple model of single-domain fer-
romagnetic nanoparticles that are characterized by the 
magnetic moment  of a constant length 
. It is assumed that the rotational dy-
namics of the vector m is described by the following 
stochastic LL equation:  
 
  (2.1) 
 
where γ(>0) is the gyromagnetic ratio, α(>0) is the 
damping parameter, the cross denotes the vector prod-
uct, is the effective magnetic field act-
ing on the magnetic moment, W is the magnetic energy 
of a particle, and  is a random magnetic field. 
Introducing the dimensionless time  and ener-
gy  (  is a characteristic magnetic field), 
the LL equation (2.1) in spherical coordinates can be 
written as a system of two equations 
 
  (2.2) 
 
Here,  and  are the polar and azimuth-
al angles of m, respectively, 
 
  (2.3) 
 
,   
, and   with 
 being the Cartesian coordinates of the random mag-
netic field h. 
We consider the coordinates  to be independent 
and identically distributed white noises. Therefore, 
since these noises are multiplicative, the system of 
equations (2.2) should be defined more precisely. To this 
end, we interpret Eqs. (2.2) in the Ito sense [13] (see 
also Refs. [5,6]). In general, to find the statistical prop-
erties of the increments  ( ), it is necessary to 
solve these equations under condition that the statisti-
cal characteristics of  are fixed. However, in order to 
 S.I. DENISOV, O.O. BONDAR, V.V. REVA PROC. NAP 1, 04MFPN15 (2012) 
 
 
04MFPN15-2 
qualitatively characterize the role of non-Gaussian 
noises in the dynamics of m, we assume that the statis-
tical properties of  are prescribed. According to the 
definitions [see formulas after Eqs. (2.3)], from the con-
dition  (the angular brackets denote an aver-
age over the realizations of ) one gets  and, 
if  (  and 
, then 
 (p  and 
. These properties permit us to consider 
the increments  as independent and identically dis-
tributed variables with zero mean and represent them 
in the form  
 
 , (2.4) 
 
where  are two white noises with identical statisti-
cal characteristics. It should be emphasized that such 
interpretation of noises in Eqs. (2.2) differs from the 
original one and the stochastic dynamics of m is also 
different. Nevertheless, we exploit this toy model to get 
more insight into the effects of non-Gaussian noises. 
In this work we study the case when  (we omit 
the index p) is Poisson white noise, i.e., a random se-
quence of delta pulses defined as 
 
 . (2.5) 
 
Here,  are independent zero-mean random variables 
with the same probability density q(z), δ(τ) is the δ 
function,  is the Poisson counting process character-
ized by the probability  (  is 
the rate of the process) that  events occur with-
in the interval (0, τ], and  are the event times. For this 
noise the probability density  that the increment 
 at  equals  is given by [14] 
 
 . (2.6) 
 
According to this result, the probability per unit 
time that  exceeds any given value is, in general, 
non-zero in the limit . As a consequence, the 
dynamics of the magnetic moment m described by 
Eqs. (2.2) is discontinuous. One more feature of these 
equations is that , while the polar and 
azimuthal angles  and  must belong to the in-
tervals  and , respectively, for all . There-
fore, to find the connection between these angles and 
the increments  and , we use the relation 
, which yields 
 
  (2.7) 
and 
  (2.8) 
 
Here, the whole numbers  and  are chosen for given 
 and  from the conditions  and 
. These numbers count the number of 
rotations of m in the corresponding planes. Thus, 
Eqs. (2.2) driven by Poisson white noise are now com-
pletely defined and we are ready to derive the corre-
sponding generalized FP equation. 
3. GENERALIZED FOKKER-PLANCK EQUA-
TION  
 
The probability density  that  
and  is defined in the standard way 
 
 . (3.1) 
 
Denoting , from the definition (3.1) 
and conditions (2.7) and (2.8) one obtains 
 
  (3.2) 
 
The average in Eq. (3.2) is equivalent to the average 
over the variables  and . Taking into 
account that these two groups of variables are inde-
pendent of each other and using Eqs. (2.2), we find 
 
   
 
   
 
  (3.3) 
 
   
 
 ,  
 
where . Finally, using Eq. (2.6) and 
the standard definition , 
after straightforward calculations we arrive at the fol-
lowing generalized FP equation: 
 
   
 
 +   
 
 +  (3.4) 
 
+   
 
 .  
 
The solution of this integro-differential equation must 
be normalized, i.e., , and satis-
fy the initial condition . Another 
form of this equation, which can be useful for studying 
its properties, follows directly from Eq. (3.4) by using 
the Poisson summation formula. 
 
4. CONCLUSIONS 
 
We have introduced a toy model for studying the ef-
fects of Poisson white noise in the rotational dynamics 
of the nanoparticle magnetic moment. We have shown 
that the dynamics is discontinuous and have developed 
an approach to account for the manifold rotations of the 
magnetic moment under pulses of the noise. Finally, 
using this approach, we have derived the corresponding 
generalized Fokker-Planck equation. 
 
ACKNOWLEGMENTS 
 
The authors are grateful to the Ministry of Educa-
tion and Science of Ukraine for the financial support. 
 GENERALIZED FOKKER-PLANCK EQUATION FOR THE NANOPARTICLE… PROC. NAP 1, 04MFPN15 (2012) 
 
 
 04MFPN15-3  
REFERENCES 
 
1. J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98, 
283 (1997). 
2. C.A. Ross, Annu.Rev. Mater. Res. 31, 203 (2001). 
3. Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, J. 
Phys. D: Appl. Phys. 36, R167 (2003). 
4. W.F. Brown Jr., Phys. Rev. 130, 1677 (1963). 
5. H. Risken, The Fokker-Planck Equation: Methods of Solu-
tion and Applications (Berlin: Springer-Verlag 1989). 
6. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin 
Equation (Singapore, WorldScientific, 2004). 
7. S.I. Denisov, K.N. Trohidou, Phys. Rev. B 64, 184433 
(2001). 
8. S.I. Denisov, T.V. Lyutyy, K.N. Trohidou, Phys. Rev. B 67, 
014411 (2003). 
9. S.I. Denisov, T.V. Lyutyy, P. Hänggi, K.N. Trohidou, Phys. 
Rev. B 74, 104406 (2006). 
10. S.I. Denisov, T.V. Lyutyy, P. Hänggi, Phys. Rev. Lett. 97, 
227202 (2006). 
11. S.I. Denisov, K. Sakmann, P. Talkner, P. Hänggi, Eu-
rophys. Lett. 76, 1001 (2006). 
12. S.I. Denisov, A.Yu. Polyakov, T.V. Lyutyy, Phys. Rev. B 
84, 174410 (2011). 
13. K. Ito, Nagoya Math. J. 1, 35 (1950) 
14. S.I. Denisov, W. Horsthemke, P. Hänggi, Eur. Phys. J. B 
68, 567 (2009). 
 


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Generalized Fokker-Planck Equation for the Nanoparticle Magnetic Moment Driven by Poisson White Noise

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