18 Time Flow in Word

Language and Spacetime

Time Flow in Word

For KOHARI Akihiro and His Time

TANAKA Akio

1

Word is expressed by phase space in the situation on dynamical system.

Phase space M has a point x.

Continuous time t is expressed by group R that consists of real number.

Homomorphism φ^t(x)expresses the system.

φ^s+t(x) =φ^s (φ^t(x)), φ⁰ (x) = x

2

In thermodynamics, equilibrium is presented.

Equilibrium situation is expressed by (U, V, n).

U is energy in the system. V is volume. n is material.

Here new equilibrium is expressed by (U’, V’, n).

3

Now a category is presented.

(U, V, n) →^f (U’, V’, n)

f is function of entropy.

Here (U, v, n) is expressed by P. (U’, V’, n) is expressed by P’.

The situation of equilibrium is expressed by the following.

f : P → P’

4

The situation of equilibrium is rewritten by phase space.

Each situation of equilibrium is expressed by a point x.

After time t, point x and x’ is presented by the following.

φ : φ^s(x) →φ^s+t(x)

5

Now equilibrium system is expressed by {φ^t} that is called flow.

Here phase space M can take tangent vector X.

Point x has tangent vector X_x = dφ^t(x)/dt_t=0

The flow has solution as equation φ^t(x)/dt = X_φ^t_(x)

6

Here set {φ^t(x) | t∈R } is presented. The set is called orbit.

7

In word, meaning is expressed by flow and time is expressed by orbit.

Tokyo May 3, 2007

Sekinan Research Field of Language

www.sekinan.org