On the Speed of Heat Waves

On the Speed of Heat Wave Mihály Makai [email protected]

Contents • • • • • •

Formulation of the problem: infinite speed? Local thermal equlibrium (LTE) hypothesis Balance equation Phenomenological balance Speed of heat wave Application in plasma transport

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M. Makai: On the Speed of Heat Wave ICTT22

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1. The Problem • Diffusion and heat conductance: infinite speed • In reactor physics: diffusion equation or telegrapher’s equation (Wigner) • The motivation of Zwanzig: memory effects. Let αi stand for the deviation of the ith state variable from its equilibrium force, then dα i (1) = ∑ Lik Fk (α1 ,..., α n ) dt k where Fk is a thermodynamic force, Lik-transport coefficent 9/13/2011


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1. The problem (cont.-1) ”According to Eq. (1), the response of a system to an applied force is simultaneous with the application of the force. As a general rule, such simultaneity in a macroscopic theory turns out to be an approximation to a casual behavior, where the response to a force comes after the application of the force.”

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1. The Problem (cont.-2) • Joseph and Preciosi in connection with heat conductance and diffusion: ”Two problems are the source of this stream: the problem of infinite wave speed and the problem of second sound.”

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1. The Problem (cont.-3) • How to estimate the speed of a heat wave in out-of-equilibrium state? – Local thermal equilibriumlimitations – Balance equationsfinite speed – Onsager’s linear modelfinite speed can be obtained

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2. Local Thermal Equilibrium (LTE) Consider a Σ out of equilibrium but its parts be in equilibrium. Then, there are such small volumes in Σ that in each volume p, T exist. Then the first law of thermodynamics holds in each small volume:

dU (r ) = T (r )dS (r ) − P(r )dV + µdM Such a small volume is „infinitesimal” but large enough to be in equilibrium (large number of collisions!).

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2. LTE (cont.-1) When the number of particles is constant, then dU (r ) = T (r )dS (r ) − P(r )dV (r )

The first law applies in each subsystem thus and U(r) and S(r) are not independent functionsthe Wronskian is zero: ∇U ∇S 9/13/2011

∂ tU =0 ∂t S


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2. LTE (cont.-2) S = ∫ ln f (r, v, t ) f (r, v, t )d 3 v 1 U = ∫ mv 2 f (r, v, t )d 3 v 2 C(r, t )∇f (r, v, t ) = ∂ t f (r, v, t )



A hyperbolic equation with one characteristics:

dr =C dt In LTE, any change in the distribution function has a limited speed.

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3. Balance equations Balance equations are generally applicable in near equilibrium state. Basic variables are the extensive X = ( X 1 ,..., X n )

quantities per unit mass and the currents are written as J(X1,...,Xn). Component index k

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ρk ρ k = ck ρ ; v = ∑ v k k ρ


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3. Balance equations (cont.-2) Mass balance: J mk ,o = ρ k v

∂ck ρ + ∇J mk ,o + ∇J mk ,e = 0 ∂t

J mk ,e = ρ k ( v − v k )

Equation of motion Energy

∂ρe + ∇J e = 0 ∂t

J ep

∂ρv + Div( ρvv + + P) = ∑ ρ k Fk ∂t k

e = ρv 2 / 2 +ψ + u kinetic

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J op

potential


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3. Balance equation (cont.-3) J = ρev + P v + ∑ψ J + J e

k

m ke

q e

k

J ee = P v + ∑ψ k J mke + J eq k

The first law connects work, heat and internal energy. The heat balance:

dq ρ + ∇J eq = 0 dt

µ ds ∇J eq − ∇∑ k J mke = qs ρ + dt T k T

 1 1 q 1  µk  m  qs = − 2 J e ∇T − ∑ J k ,e  T∇  − Fk  − P : Gradv T T k  T  T  We need a balance equation with extensives only: equation of state

P = P( ρ , T ) 9/13/2011


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3. Balance equations (cont.-4) The structure of balance equation is d Xρ + ∇J ( X ) = q ( X ) dt

∂J ik ( X ) ∂J ik ∂X m =∑ ∂x m ∂X m ∂x ∂X ∂X ∂X ∂X + Mx +My + Mz =q ∂x ∂y ∂z ∂t

When Mx,My, and Mz commute there exists a common eigenvector set X0 and we have to solve 9/13/2011


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3. Balance equations (cont.-5) We have to solve

∂X ∂X ∂X ∂X + Mx +My + Mz = X 0  f (t , r ) ∂t ∂x ∂y ∂z The solution is:

t

X  (t , r ) = X  ∫ f (t ' , r + v  ⋅ (t '−t ))dt '+ X  Φ (r − v  t ) t0

Here v =(vx,vy,vz) is an eigenvalue of the „M” matrices. That solution propagates with finite speed.

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4. Phenomenological balance J( X ) = J e ( X ) + J o ( X ) Currents depend on (t,r). Conductive current is even in t, convective current is odd in t.

J e = Λ∇Y ( X ) = ΛN∇ X J o ( X ) = vρ X

∂Yi N ij = ∂X j

∂Xρ + vρ∇ X + ΛN∆ X = q ∂t This is the telegrapher’s equation, has wave front and finite speed.

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5. Speed of Heat Wave Consider the internal energy: u=cvT c c  1 ∇  = ∇ v  = − v2 ∇u u T  u

cv ∂u   ρ + ∇  Lqq 2 ∇u  = 0 ∂t  u 

cv Lqq 2 = v u u

∂u ρ + ∇ v u ∇u = 0 ∂t ∇∇u  1 2 ∇J qe = − Lqq cv  3 (∇u ) − 2  u  u

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Finite speed!!! To get the Infinte speed:

cv J eq = Lqq 2 ∇u u


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5. Speed of heat wave Heat conduction: we start from the balance of the internal energy. Let v=0, ψ=0, then energy=u, the energy balance is (slide 12) : ρ

∂u + ∇J eq = 0 ∂t

Conductive current is a linear expression of gradients of intensives (now only u): cv  cv  1 1 q = ∇ = − ∇u ∇     J e = Lqq ∇  2 u v T  u T

 

u=cT

Substitute that into the balance to get

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5. Speed of heat wave (cont.-1) cv ∂u   ρ + ∇  Lqq 2 ∇u  = 0 ∂t  u  New notation:

vu =

Lqq cv u2

cv J = Lqq 2 ∇u = v u ∇u u q e

and with them the balance takes the form of

ρ

∂u + ∇ ( v u ∇u ) = 0 ∂t

Note that |vu| is fiinite, so the heat conduction is not instantaneous! 9/13/2011


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5. Speed of heat wave-(cont.-2) How to get the tradational heat equation? Substituting here Jeq we get:

∂u ∇∇u  1 2 ρ = − Lqq cv  3 (∇u ) − 2  u  ∂t u

Neglecting the first term, we arrive at the traditional heat equation:

ρ

∂u = λ∇∇T ∂t

That equation accords with Onsager’s principle: higher than linear gradients are neglected.

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6. Plasma stability Plasma is a strongly non-equilibrium system. Its description is based on the balance equation. 7 characteristic speeds are obtained. The steps: 1, material balance 2, momentum balance 3, energy balance 4, Maxwell equations (for B) and divB=0 7 equations 9/13/2011


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Plasma stability-(cont.-1) The balance equation takes the form: ∂X + M ( X )∇ X = q ∂t Here

(

M ( X ) = M x ( X ), M y ( X ), M z ( X )

)

For simplicity’s sake, assume that the plasma is isotropic, and Mx,My,Mz commute. Their eigenvalue problem is

b( X ) M ( X ) = β ( X )b( X ) eigenvector

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eigenvalue of dimension velocity!!! One eigenvalue for each extensive! M. Makai: On the Speed of Heat Wave ICTT22

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Plasma stability-(cont.-2) Our plasma equations: ∂ρ + ∇( ρv ) = 0 ∂t

1   ∂v (∇ × Β )× B ρ  + ( v∇) v  = −∇P + 4π   ∂t

1 2 ∂ 1 2 B  + ∇q = 0  ρv + ρe + 8π ∂t  2 

∂B = ∇ × ( v × B) ∂t

1 P 1 q = ρv v 2 + ρe +  + B × ( v × B) ρ  4π 2

∇B = 0

8 equations for B,v,ρ,e.

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Plasma stability-(cont.-3) The plasma extensive vector:

U = (ρ , ρv x , ρv y , ρv z , ρe, Bx , B y , Bz )

+

We have to solve

∂U ∂ F x ∂ F y ∂ F z + + + =0 ∂t ∂x ∂y ∂z

Where the „F”s are the extensive current components

∂ F ∂ F ∂U = ∂x ∂U ∂x

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These are the „M” matrices on slide 13 


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Plasma stability-(cont.-4) The eigenvalues of „M”matrices give the available speeds in the plasma: v x − c f , v x − bx , v x − cs , v x , v x + c f , v x + bx , v x + cs Bi bx = ; c 4 − (a 2 + b 2 )c 2 + a 2bx2 = 0 4πρ Not detailed here Alfven waves: vb,

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magnetoacoustic waves: vcs; vcf


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Concluding remarks • Diffusion equation is only an approximation, e.g. In reactor physics the correct equation is the telegrapher’s equation; • From the underlying statistical physics finite speed is obtained; • The LTE hypothesis sets a limit to the speed of transport processes; • Non-equilibrium statistical physics predicts finite speed even in far from equilibrium states. • In practical calculations the diffusion is a good approximation. 9/13/2011


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