Energy Distance Theory / Note 2 / Heat and Diffusion

Energy Distance Theory

Note 2

Heat and Diffusion

TANAKA Akio

1 Heat equation

Time     t

Situation     x

Temperature of s

²u / x²     (k ; constant)

2 High dimensional heat equation

 = ku     (k ; constant)

 is Laplacian.

3 Diffusion equation

Time     t

Situation     x

Density of minute particles

 = div ( ku )     (k ; constant)

4 Assumption of heat equation

Assumption     k = 1

 = u

5 Initial value problem

Space     Rⁿ

Heat equation      = u

Initial time     t = 0

Temperature distribution of initial time     u₀ ( x )

Transition of temperature distribution is expressed by the next.

Initial condition  = u  x∈Rⁿ , t > 0 )

Initial value     x, 0 ) = u₀ ( x )  (x∈Rⁿ )

The upper two formulas are called initial value problem.

6 Delta function

(i) δ (x) = 0

(ii) dx = 1

7 Fundamental solution of initial value problem

Function     U ( x, y, t )

  = _xU

lim_t↘0 U ( x, y, t ) =δ (x-y)

_x is Laplacian of variable x.

8 Probability density

Particle is situated by the next.

t = 0, probability 1, point y

Probability density of the particle that has Brownian motion over x- axis, time t and point x     U ( x, y, t )

9 Heat kernel

U ( x, y, t ) = K ( x-y, t )

Function K ( x, t ) is called heat kernel.

10 Hausdorff dimension

Arbitrary figure in space Rⁿ     S

Sequence of n-dimensional sphere     B₁, B₂, B₃, …    

S is covered by the sequence B_k that diameter is below δ.

H_α, δ( S ) : = inf _{diam ( Bk ) <δ} (diam(B_k))^α

H_α( S ) : = lim_k→0 H_α, δ( S )

H_α( S ) is called figure S’s α dimensional Hausdorff outer measure.

To be continued

Tokyo September 15

Sekinan Research Field of Language