Energy of Language
For ZHANG Taiyan and Wenshi 1908
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 ri∈Λ
Momentum of particle pi∈R3
State at a moment γ = (r1, …, rN, p1, …, pN)
Set of state γ PΛ, N ≃ΛN ×R3N⊂R6N
PΛ, N is called phase space.
2
Volume V
Particles n- mol
Energy U
Parameter space E
Point of E ( U, V, n )
3
Subspace PΛ, N ( U )
Volume of PΛ, N ( U ) WΛ, N ( U )
4
Adiabatic operation ( U, V, n ) → ( U’, V’, n’ )
Starting state of γ∈PΛ, N
Ending state of γ∈PΛ’, N
Map of time development f
5
Volume of PΛ’, N ( U’ ) WΛ’, N ( U’ )
Volume of f (PΛ, N ( U ) ) is equivalent to WΛ, N ( U ).
f (PΛ, N ( U ) ) is subspace of PΛ’, N ( U’ )
WΛ, N ( U ) ≤ WΛ’, 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 WΛ, N ( U ) WΛ’, N’ ( U’ )
Operation of logarithm of equilibrium state at phase space S ( U, V, n ) = k log WΛ, N ( U ) , (k ; arbitrary constant)
7
Phase space 2n- dimension
Differential 2-form ω
Local coordinate qi, pi
ω = ∑ni=1d qi, ∧dpi
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, …, qn, p1, …, pn )
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]
To be continued
Tokyo July 24, 2008
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