HSGCMC near the critical point.

• Overview
  

  Hamiltonian scaling grand canonical Monte Carlo was used to simulate twenty diffenent parameter sets. Perturbations were made about a reference fluid to the parameters e, s, and q. Calclations were completed for the reference fluid as well as; e = ± D _e, e = ± 2D _e, s = ± D _s, s = ± 2D _s, q = ± D_q, and q = ± 2D_q. The parameters D_e, D_s, and D_q were set to 8, 0.012, 0.005 respectively. In addition, seven "cross" parameter sets were simulated. The ability to express the critical temperature and density as a taylor expansion about the reference fluid is demonstrated.
  
  

• Reference fluid
  
  

  e	s	q	R
  70.0	3.145	0.415	1.00

  
  
• Effect of changing e
  
  

  	-2De	-De	Ref.	+De	+2De
  Tc	664.9	649.9	638.9	631.3	625.5
  rc	278.4	278.2	278.6	277.8	279.1

  
  
  • Critical temperature
  • Critical density
  
  
• Effect of changing s
  
  

  	-2Ds	-Ds	Ref.	+Ds	+2Ds
  Tc	659.3	649.0	638.9	629.2	619.4
  rc	282.9	280.4	278.6	276.2	273.9

  
  
  • Critical temperature
  • Critical density
  
  
• Effect of changing q
  
  

  	-2Dq	-Dq	Ref.	+Dq	+2Dq
  Tc	605.1	621.4	638.9	656.1	674.2
  rc	278.2	278.6	278.6	279.6	278.9

  
  
  • Critical temperature
  • Critical density
  
  
• Predicting critical parameters from a taylor expansion about the reference fluid.
  
  
  • T_c = T_c,0 + a₁(e-e_o) + a₂(e-e_o)² + b₁(s-s_o) + c₁(q-q_o)
    
    T_c,0 = 638.9
    
    

    index	a	b	c
    1	-1.2173	-831.37	3459.5
    2	0.024395	-	-

    
    
  • It was observed that the critical density was only effected by s.
    r_c = r_c,0 + b₁(s-s_o)
    
    r_c,0 = 278.4
    b₁ = -183.89
    
    
  • Comparison of critical parameters.
    
    

    e	s	q	Tc, sim	Tc, te	rc, sim	rc, te
    70	3.157	0.420	646.5	646.2	275.4	276.2
    70	3.133	0.410	632.0	631.5	280.9	280.6
    70	3.121	0.405	624.7	624.2	282.0	282.8
    62	3.145	0.410	632.2	632.9	277.6	278.4
    78	3.145	0.420	648.0	648.0	278.0	278.4
    62	3.157	0.415	639.8	640.2	275.1	276.2
    78	3.133	0.415	640.4	640.7	280.2	280.6

    
    

    T_c, sim - Critical temperature found from a mixed-field finite-size scaling analysis with f(N,E) histograms collected during a Hamiltonian scaling grand canonical run.

    T_c, te - Critical temperature predicted from the taylor expansion above.

    r_c, sim - Same as for T_c, sim

    r_c, te - Same as for T_c, te