Stochastic Meaning Theory 4 Energy of Language

Stochastic Meaning Theory 4

Energy of Language

For ZHANG Taiyan and Wenshi 1908

TANAKA Akio

1

Domain     Λ∈R³

Substantial particles     N-number m-mass 

Particles are assumed to Newton dynamics.

Place coordinate of particle i in N-number particles     r_i∈Λ

Momentum of particle     p_i∈R³

State at a moment     γ = (r₁, …, r_N, p₁, …, p_N)

Set of state γ     P_{Λ, N} ≃Λ^N ×R^3N⊂R^6N

P_{Λ, N} is called phase space.

2

Volume     V

Particles     n- mol

Energy     U  

Parameter space     E

Point of E     ( U, V, n )

3

Subspace     P_{Λ, N} ( U )

Volume of P_{Λ, N} ( U )     W_{Λ, N} ( U )

4

Adiabatic operation      ( U, V, n ) →  ( U’, V’, n’ )

Starting state of γ∈P_{Λ, N}

Ending state of γ∈P_{Λ’, N}

Map of time development    f

5

Volume of P_{Λ’, N} ( U’ )     W_{Λ’, N} ( U’ )

Volume of f (P_{Λ, N} ( U ) ) is equivalent to W_{Λ, N} ( U ).

f (P_{Λ, N} ( U ) ) is subspace of P_{Λ’, N} ( U’ )

W_{Λ, N} ( U ) ≤ W_{Λ’, N} ( U’ )

6

Equilibrium state     ( U, V, n )

Another equilibrium state       ( U’, V’, n’ )

Two volume of equilibrium states are seemed to be one state at phase space     W_{Λ, N} ( U ) W_{Λ’, N’} ( U’ )

Operation of logarithm of equilibrium state at phase space     S ( U, V, n ) = k log W_{Λ, N} ( U ) , (k ; arbitrary constant)

7

Phase space     2n- dimension

Differential 2-form    ω

Local coordinate     q_i, p_i

ω = ∑ⁿ_i=1d q_i, ∧dp_i

ω is called symplectic form.

2n- dimensional manifold     M

Pair    (M, ω)

(M, ω) that satisfies the next is called symplectic manifold.

(i) dω = 0

(ii) ωⁿ ≠0

Phase space is expressed by symplectic manifold.

8

Hamiltonian system

Coordinate    ( q, p ) = (q₁, …, q_n, p₁, …, p_n )

Phase space     R²ⁿ

C¹ class function     H = (q, p, t )

 = ( 1≤i ≤n )

 = ( 1≤i ≤n )

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