Time-dependent attractive thermal quantum force upon a Brownian free particle in the large friction regime
ABSTRACT We quantize the Brownian motion undergone by a free particle in the absence of inertial force (the so-called large friction regime) as described by the diffusion equation early found out by Einstein in 1905. Accordingly, we are able to come up with a time-dependent attractive quantum force F(t) that acts upon the Brownian free particle as a result of quantum-mechanical thermal fluctuations of a heat bath consisting of a set of quantum harmonic oscillators having the same oscillation frequency /omega in thermodynamic equilibrium at temperature T. More specifically, at zero temperature we predict that the zero-point force is given by F^((T=0)) (t)=-[\omega/(1+2\omegat)^(3/2)] \sqrt(\gamma/2), where \gamma is the friction constant with dimensions of mass per time and /eta the Planck constant divided by 2\pi. For evolution times t~1/\omega, \omega~0^14 Hz, /gamma~10^(-10) kg/s, and /eta~10^(-34) m^2 kg/s, we find out F^((T=0)) ~10^(-8) N, which exhibits the same magnitude order as the Casimir electromagnetic quantum force, for instance. Thus, we reckon that novel quantum effects arising from our concept of time-dependent thermal quantum force F(t) may be borne out by some experimental set-up in nanotechnology. Comment: Submitted.
Time Time- -dependent attractive thermal q dependent attractive thermal quantum
force upon a Brownian free particle in the force upon a Brownian free particle in the
large friction regime large friction regime
A. O. Bolivar*
April 4 th 2010
Time Time- -dependent attractive tdependent attractive thermal q
for force upon a ce upon a Brownian Brownian free
large friction regime large friction regime
free particle in the particle in the
A. O. Bolivar*
Instituto Mário Schönberg de Física-Matemática-Filosofia, Ceilândia, Caixa Postal
7316, 72225-971, D.F, Brazil
We quantize the Brownian motion undergone by a free particle in the
absence of inertial force (the so-called large friction regime) as described by the
diffusion equation early found out by Einstein in 1905. Accordingly, we are able to
come up with a time-dependent attractive quantum force ℱ ? that acts upon the
Brownian free particle as a result of quantum-mechanical thermal fluctuations of a
heat bath consisting of a set of quantum harmonic oscillators having the same
oscillation frequency ? in thermodynamic equilibrium at temperature ?. More
specifically, at zero temperature we predict that the zero-point force is given by
ℱ ?=0 ? = −
1 + 2?? 3/2 ?ħ
where ? is the friction constant with dimensions of mass per time and ħ the Planck
constant divided by 2?. For evolution times ?~1/?, ?~1014Hz, ?~10−10kg/s, and
ħ~10−34m2kg/s, we find out ℱ ?=0 ~10−8N, which exhibits the same magnitude
order as the Casimir electromagnetic quantum force, for instance. Thus, we reckon
that novel quantum effects arising from our concept of time-dependent thermal
quantum force ℱ ? may be borne out by some experimental set-up in
PACS numbers: 05.40.-a; 05.40. Jc; 05.60. Gg
Key words: Quantum Brownian motion; strong friction limit
phenomena in physics, chemistry, astronomy, electrical engineering, as well as in
mathematics [1—8]. Historically, Einstein  came up with a mathematical
description of Brownian motion in terms of the time evolution of the probability
distribution function ? ?,? for a free particle immersed in an environment (the
so-called diffusion equation)
The centenary theory of Brownian motion has been applied to a plethora of
= ? ∞ ?2? ?,?
where ? is the position of the Brownian particle as inertial forces upon it may be
negligible (the large friction domain). The diffusion constant ? ∞ , calculated at
the long time regime, with dimensions of ??????2× ????−1 may be written down
? ∞ =ℰ ∞
where the steady quantity ℰ ∞ having dimensions of energy is termed the
diffusion energy responsible for the Brownian motion, whereas the damping
constant ? exhibiting dimensions of mass per time is related to mechanical
properties inherent in the interaction between the Brownian massive particle and
the environmental particles.
Brownian particle, can be identified with the thermal energy ??? coming from the
heat bath in thermodynamic equilibrium at temperature ?, i.e.,
On the condition that the diffusion energy ℰ ∞ , which comes from the
ℰ ∞ = ???, (3)
we readily obtain the Einstein’s fluctuation-dissipation relation 
? ∞ =???
The thermodynamic constant ?? is dubbed the Boltzmann constant. Assumption
(3) is a form of expressing the principle of energy conservation.
On the basis of the diffusion equation (1) and the diffusion constant (4)
Einstein  derived the mean square displacement as
∆? ? = 2???
which is the physically measurable quantity in the theory of Brownian motion. For
? = 1s, ??~10−23m2kgs−2K−1, ?~300K, and ?~10−9kgs−1, Einstein obtained a
value of the order of ∆?~?m.
It is worth noticing that no diffusive motion, ? ∞ = 0, and no fluctuation,
∆? ? = 0, are predicted at zero temperature. Yet, as far as quantum effects are
concerned, both the Brownian dynamics (1) and the heat bath’s thermal energy
(3) must depend on the quantum action ħ, the Planck’s constant divided by 2?.
Therefore, the diffusion energy (3) and in consequence both the constant diffusion
(4) and the mean square displacement (5) cannot vanish at the zero-temperature
realm on account of the existence of the zero-point energy.
Although the environment can be quantum-statistically treated as a heat
bath consisting of a set of harmonic oscillators in thermal equilibrium at
temperature ? , the quantization process of the Brownian dynamics (1) holds
still a terra incognita. We reckon that inquiring into this outstanding physical
problem can shed some light on the quantum nature of Brownian motion thereby
implying ongoing advances in nanotechnology, for example.
The aim of the present paper is then to take up the physical-mathematical
problem of quantizing the diffusion equation (1) reckoning with the quantum
nature of the heat bath. To this end, we start from it, with (2), at points ?1 and ?2,
Solutions ? ?1,? and ? ?2,? are deemed to be associated with the diffusion
energy ℰ ∞ = 2ℰ(∞). By multiplying (6) by ? ?2,? and (7) by ? ?1,? and
adding the resulting equations, we obtain
where ? ?1,?2,? = ? ?1,? ? ?2,? = ? ?1,? ? ?2,? .
By performing the following change of variables into configuration space,
?1,?2 ⟼ ?,?ħ ,
?1= ? −?ħ
?2= ? +?ħ
the classical equation (8) changes into the quantum equation of motion
where we have replaced the solution ? ?1,?2,? with ? ?,?,? and employed the
subscript notation in ℰħ ∞ to disclose the quantum nature of the diffusion energy
which turns out to be now expressed in terms of the Planck constant ħ. The
variable ? displays dimensions of inverse of linear momentum.
The geometric meaning of the quantization conditions (9) and (10) is
related to the existence of a minimal distance between the points ?1 and ?2 due to
the quantum nature of space, ?2− ?1= ?ħ, such that in the classical limit ħ → 0,
physically interpreted as ?ħ ≪ ?2− ?1 , the result ?2= ?1= ? can be readily
By making use of the Fourier transform
? ?,?,? =
2? ? ?,?,? ??????,
which changes the variables from quantum configuration space (?,?ħ) onto
quantum phase space ?,?;ħ , the quantum dynamics (11) turns out to be written
?2? ?,?,? . (13)
Because the exponential factor ???? in (12) is to be a dimensionless term, it follows
that ? is to have dimensions of linear momentum. Thus both quantum equations of
motion (11) and (13) arise from our method of quantizing the Brownian dynamics
We now assume that the environment is made up of a set of harmonic
oscillators having the same oscillation frequency ?, so that the mean energy of this
quantum heat bath after attaining the thermodynamic equilibrium at temperature
? is given by the Boltzmann—Maxwell statistics 
2??? , (14)
where the ħ-dependent energy, ?ħ/2, corresponds to the zero-point energy
inherent in the quantum heat bath at zero temperature and ??? the classical
thermal energy of the quantum heat bath at high temperatures ? ≫ ?ħ/2??.
On the condition that the Brownian particle’s quantum diffusion energy
ℰħ ∞ can be identified with the heat bath’s quantum thermal energy ?, we obtain
ℰħ ∞ =?ħ
2??? , (15)
from which the quantum diffusion coefficient can be derived as
?ħ ∞ =ℰħ ∞
2??? . (16)
Our quantum phase-space diffusion equation (13) may be solved starting
from the non-thermal initial condition
? ?,?,? = 0 =
leading to the distribution ? ?,? = ? ? ? ? in the classical limit, ħ → 0. The
constant ? has dimensions of time per mass, hence we may set ? = 1/?. The
quantum steady probability distribution function (17) generates the fluctuations
∆? 0 = ?ħ/2 and ∆?(0) = ħ/2?, which comply with the Heisenberg
indeterminacy principle ∆?(0) ∆?(0) = ħ/2.
The time-dependent solution to (13), with (15), reads
? ?,?,? =
4?ħ ? , (18)
?ħ ? =
Solution (18) leads then to both the quantum mean square displacement
∆? ? = ħ
2? 1 + 2?? coth
and the quantum mean square momentum
∆? ? = ?ħ
1 + 2?? coth ?ħ
which in turn do fulfill the Heisenberg constraint: ∆? ? ∆?(?) = ħ/2.
The quantum mean square displacement (19) gives rise to the quantum
velocity ? ? = ?∆? ? /??, while the quantum mean square momentum (20)
generates the time-dependent attractive thermal quantum force ℱ ? = ?∆? ? /??
ℱ ? = − ?ħ
1 + 2??coth ?ħ
ranging from ℱ ? = 0 = − ?ħ/2?coth ?ħ/2??? to ℱ ∞ = 0.
From the Heisenberg relation ∆? ? ∆?(?) = ħ/2 it is readily to show that
the quantities ℱ ? and ?(?) are connected via the relationship ℱ ? =
− ∆? ? /∆? ? ?(?). From the physical viewpoint the attractive quantum force
(21) is exerted upon the Brownian free particle as a result of quantum-mechanical
thermal fluctuations of the heat bath as well as the quantum Brownian dynamics
The thermal quantum force (21) at zero temperature reads
ℱ ?=0 ? = − ?ħ
1 + 2?? 3/2. (22)
For evolution times ?~1/?, ?~1014 Hz, ?~10−10kg/s, and ħ~10−34m2kg/s, we
find out the following magnitude: ℱ ?=0 ~10−8N. Under these conditions the
mean square displacement (19) is of the order of ∆? ?=0 ~10−12m. Thereby, it is
worth underscoring that our quantum force (21) may exhibit the same magnitude
order as the Casimir force [10,11], for instance. Consequently, it may be borne out
The high temperature regime should be physically interpreted as the
temperature ? is deemed to be too large in comparison with the quantum
temperature ?q= ?ħ/2?? , i.e., ? ≫ ?ħ/2??, such that coth ?ħ/2??? ~2???/
?ħ. Thus, the mean square displacement (19) becomes
∆? ? = ħ
where the ħ-dependence comes from the quantum dynamics (13) starting from
the initial condition (17) while the ħ-independent term is due to the classical
nature of the heat bath. On the other hand, our thermal quantum force (21) at high
temperature turns out to be given by
ℱ ? = −
ħ + 4???? 3/2. (24)
For ?~10−10kg/s at ?~103K, and ? = 0, Eq. (23) and Eq. (24) lead to
∆?(0)~10−12m and ℱ ? = 0 ~10−8N, respectively.
The classical limit of the Brownian dynamics should be interpreted as the
evolution time scale ? is considered too large in comparison with the quantum time
scale ?q= ?−1= ħ/4???, i.e., ? ≫ ħ/4???. At 100 K, ?q~10−13s, for instance. So
Eq. (23) reduces to the Einstein’s classical mean square displacement 5 . Taking
into account the Einstein’s particle  characterized by ?~10−10kg/s at ? = 1s,
Eq.(24) leads to the magnitude ℱ ? = 1s ~10−30N. In other words, our quantum
force (21) virtually vanishes in the classical realm characterized by ? ≫ ?q and
? ≫ ?q.
In summary, in this paper we have examined the quantum Brownian
motion of a free particle in the absence of inertial force (the so-called strong
friction case). Our crucial finding is the concept of time-dependent attractive
thermal quantum force (21) exerted upon the quantum Brownian free particle by
the quantum-mechanical heat bath. We reckon that this intrinsically quantum
physical effect can be experimentally confirmed since at zero temperature it may
have the same magnitude order as the Casimir force, for instance. Moreover, our
upshot (21) may play a pivotal role in quantum nanothermomechanical systems at
the low-temperature range ?~?ħ/2??, as it virtually dies out at high
I thank Professor Maria Carolina Nemes for the scientific support and
FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais) for the
*Present address: Departamento de Física, Universidade Federal de Minas
Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil.
Electronic mail: firstname.lastname@example.org.
 A. Einstein, Ann. Phys. 17
 N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed.
(Elsevier, Amsterdam, 2007).
17, 54 (1905).
 H. Risken, The Fokker—Planck Equation: Methods of Solution and Applications,
2nd ed. (Springer, Berlin, 1989).
 C. W. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry, and
the Natural Sciences, 3rd ed. (Springer, Berlin, 2004).
 W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation: with
Applications to Stochastic Problems in Physics, Chemistry and Electrical
Engineering 2nd ed. (World Scientific, Singapore, 2004).
 R. M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications (Oxford
University Press, New York, 2002).
 U. Weiss, Quantum Dissipative Systems 3rd ed. (World Scientific, Singapore,
 A. O. Bolivar, Quantum—Classical Correspondence: Dynamical Quantization
and the Classical Limit (Springer, Berlin, 2004).
 R. C. Tolman, The Principles of Statistical Mechanics, (Dover, New York, 1979).
 H. B. G. Casimir, Koninkl. Ned. Adak. Wetenschap. Proc. 51
Lamoreaux, Phys. Rev. Lett. 78 78, 5 (1996).
51, 793 (1948); S. K
 It should be noticed that our force (21) and the Casimir force are quantum
forces of distinct physical origins. The former is a time-dependent thermal
quantum force, whereas the latter is a position-dependent electromagnetic
quantum force. Yet both are attractive forces.