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**that consists of real number.***R*
Homomorphism φ

*(*^{t}*x*)expresses the system.
φ

^{s+t}(*x*) =φ^{s}*(φ**(*^{t}*x*)), φ*(*^{0 }*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) →

*(U’, V’, n)*^{f }*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**} that is called*^{t}**flow**.
Here phase space

*M*can take tangent vector*X.*
Point

*x*has tangent vector*X*_{x }=*d**φ**(*^{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*∈**} is presented. The set is called***R***orbit**.
7

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

**Tokyo May 3, 2007**

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