Hypernetted-chain equation

<p>In <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> the hypernetted-chain equation is a <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closure</a> relation to solve the <a href="/facts/Ornstein%25E2%2580%2593Zernike_equation/FBUedwnW">Ornstein–Zernike equation</a> which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the <a href="/facts/Radial_distribution_function/zWN3v1Cr">radial distribution function</a>. It is given by:
</p>

ln
        ⁡
        y
        (
        
          r
          
            12
          
        
        )
        =
        ln
        ⁡
        g
        (
        
          r
          
            12
          
        
        )
        +
        β
        u
        (
        
          r
          
            12
          
        
        )
        =
        ρ
        ∫
        
          [
          
            h
            (
            
              r
              
                13
              
            
            )
            −
            ln
            ⁡
            g
            (
            
              r
              
                13
              
            
            )
            −
            β
            u
            (
            
              r
              
                13
              
            
            )
          
          ]
        
        h
        (
        
          r
          
            23
          
        
        )
        
        d
        
          
            r
            
              3
            
          
        
        ,
        
      
    
    {\displaystyle \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho \int \left[h(r_{13})-\ln g(r_{13})-\beta u(r_{13})\right]h(r_{23})\,d\mathbf {r_{3}} ,\,}

<p>where 
  
    
      
        ρ
        =
        
          
            N
            V
          
        
      
    
    {\displaystyle \rho ={\frac {N}{V}}}
  
 is the <a href="/facts/Number_density/qtyDnLhz">number density</a> of molecules, 
  
    
      
        h
        (
        r
        )
        =
        g
        (
        r
        )
        −
        1
      
    
    {\displaystyle h(r)=g(r)-1}
  
,  
  
    
      
        g
        (
        r
        )
      
    
    {\displaystyle g(r)}
  
 is the <a href="/facts/Radial_distribution_function/zWN3v1Cr">radial distribution function</a>, 
  
    
      
        u
        (
        r
        )
      
    
    {\displaystyle u(r)}
  
 is the direct interaction between pairs. 
  
    
      
        β
        =
        
          
            1
            
              
                k
                
                  
                    B
                  
                
              
              T
            
          
        
      
    
    {\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}
  
 with  
  
    
      
        T
      
    
    {\displaystyle T}
  
 being the <a href="/facts/Thermodynamic_temperature/JXeJJPa6">Thermodynamic temperature</a> and 
  
    
      
        
          k
          
            
              B
            
          
        
      
    
    {\displaystyle k_{\rm {B}}}
  
 the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>.
</p>

Hypernetted-chain equation open-in-new

Hypernetted-chain equation