Transcript Generalization of Einstein`s Theory of Brownian Motion

Generalization of
Einstein’s Theory of
Brownian Motion
Mahmoud A. Melehy
University of Connecticut
Storrs, CT 06269-1157
Albert Einstein
(1879-1955)
Nobel Prize 1921
Thermal Momentum
Significance 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 Physical
definition of Chemical Potential
The Gibbs Equation
dU  TdS  pdV  µdN (5)
The Gibbs-Duhem Equation
U  TS  PV  µN (6)
1
dµ  ds  dP (7)
n
S
s ,
N
N
n
(8)
V
The Gibbs Equation
dU  TdS  pdV  µdN (5a)
The Gibbs-Duhem Equation
U  TS  PV  µN (5b)
1
P

s  u   µ (6)
T
n

1
dµ  ds  dP (7)
n
N
S
n
(8)
s ,
V
N
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 of
the 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. 231244,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
For any one constituent, the Gibbs-Duhem equation:
1
dµ  ds  dP (1)
n
Quantum mechanics allows to writing:
n  n( µ, T )
P  P( µ, T ) (2)
Therefore, (1)
1
dµ    dP(µ, T )T (3)
 n T
µ
P ( µ, T ) 
 n(µ' , T ) dµ' (4)
T

µ0 = -
Let the arbitrary value 
. Then
and (4) 
PP(( µ
T,T)
)  00, for any value of T
,
(5)
µ
PP(( µ,,TT)) 
 n[n((µ',,TT )]) ddµ'



TT
1
d s dT1 dP
dµ  ds  dP
n
n
Solving (1) for s, we get
1 1
P(
P,T )
s
s n T
n T
(6)
(1)
(7)
For the general case of conduction electrons in solids,
quantum mechanics has led to
For conduction electrons in solids, generally:


F
) ,F(
T, 
nn g g(
T 
d,T) ,T d
(8)
0
0
Here, 
) is
is the volume is
is the electron energy, gg(
energy density of quantum states, and FF is the FermiDirac function:
F  , µ
T, ,T) 
F(,
(µ)/kT
 / kT
1
1 ee
1
1
(9)

Equations (6),(8), and (9) 

 ) 
P µP(
, T,T
µ

g(

)F(

,
,T)
d
d





g

F

,
µ
'
,
T
d
dµ
'




T

(10)



0
,T
,T
Carrying out a few mathematical steps, including
integration by parts of (10), we obtain
P(,
P µ
T,T )
µ
F(' ,
, ,T)
d'
F)d
,µ
T
d  dµ
 g g(



0

0
T
,T
1 
P( ,T )
1
P
ss  n 
T
n T
(11)
(7)
Substituting (11) into (7), we get
11  PP

s s= uu 
µ 
TT  nn

provided that
u = the kinetic energy per particle
(12)
(13)
Generally, for conduction electrons in metals and
semiconductors, the energy band is considered to be
parabolic; i.e.
g =CC 

(13)
(11) and (13) 




22
3
/3/2
2 F(
P

C
,,T)
dd
 b nu
(14)
P  3 C 0  F  ,,µ
T 

µ,T
,T
3 0
which is the internal pressure that represents the time-rate
of change of momentum, associated with the thermal
motion of the particles. This is another confirmation of the
validity of Einstein’s basic postulate underlying the
Brownian motion theory.

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 interfaces 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.