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Wednesday, 7 January 2015

Stochastic Meaning Theory 4 Energy of Language For ZHANG Taiyan and Wenshi 1908 / July 24, 2008

Stochastic Meaning Theory 4

Energy of Language
For ZHANG Taiyan and Wenshi 1908

TANAKA Akio

1
Domain     ΛR3
Substantial particles     N-number m-mass 
Particles are assumed to Newton dynamics.
Place coordinate of particle i in N-number particles     riΛ
Momentum of particle     piR3
State at a moment     γ = (r1, …, rN, p1, …, pN)
Set of state γ     PΛ, N ΛN ×R3NR6N
PΛ, N is called phase space.
2
Volume     V
Particles     n- mol
Energy     U  
Parameter space     E
Point of E     UVn )
3
Subspace     PΛ, N ( U )
Volume of PΛ, N ( U )     WΛ, N ( U )
4
Adiabatic operation      UVn )   U’V’n’ )
Starting state of γPΛ, N
Ending state of γPΛ’, N
Map of time development    f
5
Volume of PΛ’, N U’ )     WΛ’, N ( U’ )
Volume of (PΛ, N ( U ) ) is equivalent to WΛ, N ( U ).
(PΛ, N ( U ) ) is subspace of PΛ’, N U’ )
WΛ, N ( U )  WΛ’, N ( U’ )
6
Equilibrium state     ( UVn )
Another equilibrium state       U’V’n’ )
Two volume of equilibrium states are seemed to be one state at phase space     WΛ, N ( U ) WΛ’, N’ ( U’ )
Operation of logarithm of equilibrium state at phase space     S ( UVn ) = k log WΛ, N ( U ) , (k ; arbitrary constant)
7
Phase space     2n- dimension
Differential 2-form    ω
Local coordinate     qipi
ω = ∑ni=1d qidpi
ω is called symplectic form.
2n- dimensional manifold     M
Pair    (Mω)
(Mω) that satisfies the next is called symplectic manifold.
(i) dω = 0
(ii) ω0
Phase space is expressed by symplectic manifold.
8
Hamiltonian system
Coordinate    ( qp ) = (q1, …, qnp1, …, pn )
Phase space     R2n
C1 class function     = (qpt )
 = ( 1n )
 = ( 1n )
9
An assumption from upper 8
H : = Sentence
q : = Place where word exists
p : = Momentum of word
t : = Time at which sentence is generated
10
Equilibrium state of sentence     H
Another equilibrium state of sentence     H
Adiabatic process of language     H  H
Entropy of language     S
H  H’  S (H )  S (H’ )

[References]
Warp Theory / Tokyo October 24, 2004
Quantum Warp Theory Warp / Tokyo December 31, 2005

To be continued
Tokyo July 24, 2008
Sekinan Research Field of Language

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