Transport coefficient

A transport coefficient 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 measures how rapidly a perturbed system returns to equilibrium.
The transport coefficients occur in <a href="/facts/Transport_phenomenon/Th5VWjpk">transport phenomenon</a> with transport laws

J
          
          
            k
          
        
        =
        
          γ
          
            k
          
        
        
          
            X
          
          
            k
          
        
      
    
    {\displaystyle \mathbf {J} _{k}=\gamma _{k}\mathbf {X} _{k}}

where:

J
          
          
            k
          
        
      
    
    {\displaystyle \mathbf {J} _{k}}
  
 is a flux of the property 
  
    
      
        k
      
    
    {\displaystyle k}

the transport coefficient 
  
    
      
        
          γ
          
            k
          
        
      
    
    {\displaystyle \gamma _{k}}
  
 of this property 
  
    
      
        k
      
    
    {\displaystyle k}

X
 
 
 k
 
 
 
 
 {\displaystyle \mathbf {X} _{k}}
 
, the gradient force which acts on the property 
 
 
 
 k
 
 
 {\displaystyle k}
 
.
Transport coefficients can be expressed via a <a href="/facts/Green%25E2%2580%2593Kubo_relations/UEjtW7zY">Green–Kubo relation</a>:

γ
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        
          ⟨
          
            
              
                
                  A
                  ˙
                
              
            
            (
            t
            )
            
              
                
                  A
                  ˙
                
              
            
            (
            0
            )
          
          ⟩
        
        
        d
        t
        ,
      
    
    {\displaystyle \gamma =\int _{0}^{\infty }\left\langle {\dot {A}}(t){\dot {A}}(0)\right\rangle \,dt,}

where 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is an observable occurring in a perturbed Hamiltonian, 
 
 
 
 ⟨
 ⋅
 ⟩
 
 
 {\displaystyle \langle \cdot \rangle }
 
 is an ensemble average and the dot above the A denotes the time derivative.
For times 
 
 
 
 t
 
 
 {\displaystyle t}
 
 that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized <a href="/facts/Einstein_relation_(kinetic_theory)/pLALxmow">Einstein relation</a>:

2
        t
        γ
        =
        
          ⟨
          
            
              |
            
            A
            (
            t
            )
            −
            A
            (
            0
            )
            
              
                |
              
              
                2
              
            
          
          ⟩
        
        .
      
    
    {\displaystyle 2t\gamma =\left\langle |A(t)-A(0)|^{2}\right\rangle .}

In general a transport coefficient is a tensor.

Transport coefficient open-in-new

Transport coefficient