Generalization of Einstein’s Theory of

Brownian Motion

Mahmoud A. MelehyUniversity of Connecticut

Storrs, CT 06269-1157

Albert Einstein(1879-1955)

Nobel Prize 1921

Thermal MomentumSignificance of Einstein’s

Postulate

• Type: translational, vibrational, and/or rotational

• H2, at 300 K, vrms = 1.93 km/s = 6,960 km/hour

• Conduction electrons in Cu, vrms = 1,570 km/s = 5.65x106 km/hour.

The Principle of Detailed Balancing

Liquid in equilibrium with its vapor. For Hg, at 0o C, nl /nv 6.3x109.

The Gibbs Equation and Physicaldefinition of Chemical Potential

The Gibbs Equation

(5) µdNpdVTdSdU

)6( µNPVTSU

(7) 1

dPn

dsdµ

,N

Ss (8)

V

Nn

The Gibbs-Duhem Equation

The Gibbs Equation

(5a) µdNpdVTdSdU

)5( bµNPVTSU

(7) 1

dPn

dsdµ

,N

Ss (8)

V

Nn

The Gibbs-Duhem Equation

(6) 1

µ

n

Pu

Ts

Thermodynamic Generalization of The Maxwell-Einstein Diffusion Force

The Principle of Detailed Balancing

Liquid and its vapor at equilibrium.

Analogy with Electric Circuits

---

-−+

+

V or L

Theory & Experiment for Ge & Si Diodes

Measurements by Sah (1962)

In Sb and Ga As Diodes

Measurements by Stocker (1961) Measurements by Rediker & Quist (1963)

Solar Cell Theory & Experiment

New Consequences ofthe First & Second Laws

Interfacial Forces, Entropy Change

Interfacial Electrification

Water Film on Glass

M. A. Melehy, Phys. Essays, vol.11, No. 3, pp. 430-443, 1998.

Water Film on Corian

M. A. Melehy, Phys. Essays, vol.11, No. 3, pp. 430-443, 1998.

Surface Charge on Corian

M.A. Melehy, Proc. 8th Int. Symp. on Particles on Surfaces, Brill Academic Publishers, Ed. K.L. Mittal, pp. 231-244, 2003.

Surface Charge on Styrofoam

Surface Charge on Mahogany Wood

Dipole-Charge Effects on Water-Glass Interfaces

J. Walker, Sci. Am., 251, (4), 144-154 (1984).

Dipole-Charge Effects on Water-Glass Interfaces

J. Walker, Sci. Am., 251, (4), 144-154 (1984).

Colored water flowing out of a teapot.

Forces Shaping Tornadoes

Tornadoes and Lightning

Dew Accumulation on Grass

M.A. Melehy, Proc. of the Eighth Int. Sym. on Particles on Surfaces, Ed. K.L. Mittal, pp. 231-244,VSP (2003), Utrecht, Boston.

Phenomenon of Rising Mist

Canadian Niagra Falls

Phenomenon of Rising Mist

Canadian Niagra Falls

Example of Conduction Electrons in Metals and Semiconductors

• Consistency of Einstein’ Theory of Brownian Motion with:

• 1. The first and second laws of

thermodynamics.

• 2. The quantum theory.

Thermal Momentum and Entropy Uniqueness

(1) 1

dPn

dsdµ

),( Tµnn (2) ),( TµPP

(3) ),(1

TTµdPn

dµT

(4) '),'(),(

µ

T dµTµnTµP

For any one constituent, the Gibbs-Duhem equation:

Quantum mechanics allows to writing:

Therefore, (1)

Let the arbitrary value 0 = - . Then

P( ,T) 0, for any value of T (5)and (4)

P(,T )

[n(,T )]T d (6)

d s dT1n

dP (1)Solving (1) for s, we get

s 1n

P(,T )T (7)

For the general case of conduction electrons in solids,quantum mechanics has led to

Let the arbitrary value 0 = - . Then

P( ,T) 0, for any value of T (5)and (4)

P(,T )

[n(,T )]T d (6)

d s dT1n

dP (1)Solving (1) for s, we get

s 1n

P(,T )T (7)

For the general case of conduction electrons in solids,quantum mechanics has led to

P( ,T) 0, for any value of T (5)and (4)

P(,T )

[n(,T )]T d (6)

d s dT1n

dP (1)Solving (1) for s, we get

s 1n

P(,T )T (7)

µ

0),( TµP

n

1

T

Ps

1

dPn

dsdµ

µ

T dµTµnTµP '),'(),(

For conduction electrons in solids, generally:

n

0g( ) F(,,T)

,Td (8)

Here, is the electron energy, g() is the volume-

energy density of quantum states, and F is the Fermi-

Dirac function:

F( , ,T ) 1

1 e ( )/kT(9)

Equations (6),(8), and (9)

P( ,T )

0g( ) F( ,,T)

,T d,T

d(10)

For conduction electrons in solids, generally:

n

0g( ) F(,,T)

,Td (8)

Here, is the electron energy, g() is the volume-

energy density of quantum states, and F is the Fermi-

Dirac function:

F( , ,T ) 1

1 e ( )/kT(9)

Equations (6),(8), and (9)

P( ,T )

0g( ) F( ,,T)

,T d,T

d(10)

n

0g( ) F(,,T)

,Td (8)

F( , ,T ) 1

1 e ( )/kT(9)

P( ,T )

0g( ) F( ,,T)

,T d,T

d(10)

0

, dTFgn T

is is g

F

kTµeTµF

/1

1,,

µ

T dµdTµFgTµP ' ,',,

Carrying out a few mathematical steps, includingintegration by parts of (10), we obtain

P( ,T )

0

0

g()dF(, ,T) ,T

d (11)

s 1n

P(,T )T (7)

Substituting (11) into (7), we get

s =1T

u Pn

(12)provided that

u = the kinetic energy per particle (13)

Carrying out a few mathematical steps, includingintegration by parts of (10), we obtain

P( ,T )

0

0

g()dF(, ,T) ,T

d (11)

s 1n

P(,T )T (7)

Substituting (11) into (7), we get

s =1T

u Pn

(12)provided that

u = the kinetic energy per particle (13)

P( ,T )

0

0

g()dF(, ,T) ,T

d (11)

s 1n

P(,T )T (7)

s =1T

u Pn

(12)provided that

µ

T dµdTµFgTµP ' ,',,

n

1

T

Ps

µ

n

Pus

T

1

Generally, for conduction electrons in metals andsemiconductors, the energy band is considered to beparabolic; i.e.

g = C (13)(11) and (13)

P 23

C

03/2 F( , ,T) ,T

d b n u (14)

which is the internal pressure that represents the time-rateof change of momentum, associated with the thermalmotion of the particles. This is another confirmation of thevalidity of Einstein’s basic postulate underlying theBrownian motion theory.

Generally, for conduction electrons in metals andsemiconductors, the energy band is considered to beparabolic; i.e.

g = C (13)(11) and (13)

P 23

C

03/2 F( , ,T) ,T

d b n u (14)

which is the internal pressure that represents the time-rateof change of momentum, associated with the thermalmotion of the particles. This is another confirmation of thevalidity of Einstein’s basic postulate underlying theBrownian motion theory.

parabolic; i.e.g = C (13)

(11) and (13)

P 23

C

03/2 F( , ,T) ,T

d b n u (14)

Cg

0

,2/3 ,,

3

2 dTµFCP Tµ

Summary and Conclusion

• Generalizing thermodynamically Einstein’s theory of Brownian motion has led to an interfacial transport theory, which, in turn led to many consequences, including:

• Revealing that the first and second laws of thermodynamics require the existence of electric charges on most surfaces, membranes and other interfaces.

• This nearly universal property of inter-faces makes it possible to readily explain

• many diverse phenomena, such as: ‘surface’ tension, capillarity, particle adhesion, the separation of charges upon phase change, atmospheric electricity, fog and cloud suspension, and even one mysterious phenomenon that has been observed since ancient times: the generation of static electricity by rubbing two different, insulating surfaces against one another. How much had this particular phenomenon been explained before is described, in the May, 1986 issue of Physics Today, by D. M. Burland, and L. B. Schein, who

have stated: "That some materials can acquire an electric charge by contact or rubbing has been known at least since the time of Thales of Miletus, around 600 B.C., and much work has been done on understanding the phenomenology of the effect, particularly in the 18th, and 19th centuries; nevertheless the underlying physics of electrostatic charging of insulators remains unclear.“

Generalizing Einstein’s theory of Brownian motion to interfacial systems has unlocked this ancient mystery, and many other ones too.

Thank you.