1 On Boltzmann’s principle, entropy S is expressed by the following.
S = kBlogW
kB is Boltzmann constant.
W is thermodynamic weight. On Newton dynamics, W is expressed by measure of set satisfied needed conditions in phase space.
2 H-theorem is presented by the following in accordance with Boltzmann’s principle.
Probability that gases has velocity v = (vx, vy, vz) is expressed by f (v, t).
H(t) = ∫f (v, t) log f (v,t) dv d H (t) / dt ≦ 0
Thermal equilibrium is expressed by the minimum of H.
3 On symmetry flow language, abbreviated to SFL, the followings are supposed.
1* Language has energy.
2* Language regulates position r = (x, y, z).
3* Language regulates momentum p = (px, py, pz).
4 SFL is supposed to satisfies Boltzmann’s principle.
5 Word of SFL has structure. The structure consists of meaning and time of beginning point abbreviated to mb and tb, meaning and time of ending point abbreviated to me and te. The succession of meaning from tb to te in word is called flow.
6 Here flow is expressed by {φt}.
Flow has equation solution when vector field X is defined.
dφt(r) / dt = Xφt(r) (1)
7 Now on oriented closed surface, Riemann metrics gij and Ricci curvature Rij is presented.
Ricci flow on manifold is expressed by the following.
d gij / dt = -2 gij (2)
Ricci flow (2) can be seen as a better form of equation (1).
8 SFL is considered by equation (1) and Ricci flow (2), especially (2) in relation with Thurston conjecture.
Tokyo May 5, 2007
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