// Partial derivative of F with respect to T; Mu_h and V held constant.
func dFdT(env *tempAll.Environment, F envFunc) (float64, error) {
ct := 0
// G gets F given Beta (allow x to vary; constant Mu_h)
G := func(Beta float64) (float64, error) {
ct += 1
// save the environment state before changing it
// (don't want one call of F to affect the next)
oD1, oBeta, oX, oMu_b := env.D1, env.Beta, env.X, env.Mu_b
env.Beta = Beta
// fix free variables
eps := 1e-9
_, err := D1F0XSolve(env, eps, eps)
if err != nil {
return 0.0, err
}
vF, err := F(env)
if err != nil {
return 0.0, err
}
// restore the environment
env.D1, env.Beta, env.X, env.Mu_b = oD1, oBeta, oX, oMu_b
return vF, nil
}
h := 1e-4
epsAbs := 1e-5
deriv, err := solve.OneDimDerivative(G, env.Beta, h, epsAbs)
//fmt.Println("dF_dT ct", ct)
return -math.Pow(env.Beta, 2.0) * deriv, err
}
// Partial derivative of Mu_h with respect to T; x and V held constant.
func dMu_hdT(env *tempAll.Environment) (float64, error) {
ct := 0
// F gets Mu_h given Beta
F := func(Beta float64) (float64, error) {
ct += 1
// save the environment state before changing it
// (don't want one call of F to affect the next)
oD1, oMu_h, oBeta, oMu_b := env.D1, env.Mu_h, env.Beta, env.Mu_b
env.Beta = Beta
// fix free variables
eps := 1e-9
_, err := D1MuF0Solve(env, eps, eps)
if err != nil {
return 0.0, err
}
Mu_h := env.Mu_h
// restore the environment
env.D1, env.Mu_h, env.Beta, env.Mu_b = oD1, oMu_h, oBeta, oMu_b
return Mu_h, nil
}
h := 1e-4
epsAbs := 1e-5
deriv, err := solve.OneDimDerivative(F, env.Beta, h, epsAbs)
//fmt.Println("MuT ct", ct)
return -math.Pow(env.Beta, 2.0) * deriv, err
}
// Solve the (D1, Mu_h, Beta) system with x and F0 fixed.
func D1MuBetaSolve(env *tempAll.Environment, epsAbs, epsRel float64) (vec.Vector, error) {
// our guess for beta should be above beta_c
if env.A == 0.0 && env.B == 0.0 {
D1, Mu_h, F0 := env.D1, env.Mu_h, env.F0
env.F0 = 0.0 // F0 is 0 at T_c
_, err := tempCrit.CritTempSolve(env, epsAbs, epsRel)
if err != nil {
return nil, err
}
fmt.Printf("%v; Tc = %f\n", env, 1.0/env.Beta)
omegaFit, err := tempCrit.OmegaFit(env, tempCrit.OmegaPlus)
if err != nil {
return nil, err
}
env.A, env.B = omegaFit[0], omegaFit[2]
env.PairCoeffsReady = true
env.Beta += 0.1
// we are at T < T_c; uncache env
env.D1, env.Mu_h, env.F0 = D1, Mu_h, F0
}
//fmt.Printf("%v; Tc = %f\n", env, 1.0 / env.Beta)
// solve low temp system for reasonable values of D1 and Mu_h first
_, err := D1MuSolve(env, epsAbs, epsRel)
if err != nil {
return nil, err
}
// solve the full low temp system
system, start := D1MuBetaSystem(env)
solution, err := solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
return solution, nil
}
func SolveNoninteracting(env *tempAll.Environment, epsAbs, epsRel float64) (vec.Vector, error) {
env.F0 = 0.0
env.Mu_h = 0.3
env.Beta = 50.0
system, start := NoninteractingSystem(env)
solution, err := solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
return solution, nil
}
// For use with solve.Iterative:
func CritTempStages(env *tempAll.Environment) ([]solve.DiffSystem, []vec.Vector, func([]vec.Vector)) {
vars0 := []string{"D1", "Mu_h"}
vars1 := []string{"Beta"}
diffD1 := tempPair.AbsErrorD1(env, vars0)
diffMu_h := tempPair.AbsErrorBeta(env, vars0)
system0 := solve.Combine([]solve.Diffable{diffD1, diffMu_h})
diffBeta := AbsErrorBeta(env, vars1)
system1 := solve.Combine([]solve.Diffable{diffBeta})
stages := []solve.DiffSystem{system0, system1}
start := []vec.Vector{[]float64{env.D1, env.Mu_h}, []float64{env.Beta}}
accept := func(x []vec.Vector) {
env.D1 = x[0][0]
env.Mu_h = x[0][1]
env.Beta = x[1][0]
}
return stages, start, accept
}
// Solve the environment under the conditions at T = T_c.
func CritTempSolve(env *tempAll.Environment, epsAbs, epsRel float64) (vec.Vector, error) {
// our guess for beta should be a bit above Beta_p
pairSystem, pairStart := tempPair.PairTempSystem(env)
_, err := solve.MultiDim(pairSystem, pairStart, epsAbs, epsRel)
if err != nil {
return nil, err
}
env.Beta += 0.1
// solve crit temp system for reasonable values of Mu and D1 first
system, start := CritTempD1MuSystem(env)
_, err = solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
// solve the full crit temp system
system, start = CritTempFullSystem(env)
solution, err := solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
return solution, nil
}
// Solve the (D1, Mu_h, Beta) system with x and Mu_b fixed.
func FlucTempSolve(env *tempAll.Environment, epsAbs, epsRel float64) (vec.Vector, error) {
// fix pair coefficients
if env.A == 0.0 && env.B == 0.0 && env.FixedPairCoeffs {
D1, Mu_h, Mu_b, Beta := env.D1, env.Mu_h, env.Mu_b, env.Beta
env.Mu_b = 0.0 // Mu_b is 0 at T_c
_, err := tempCrit.CritTempSolve(env, epsAbs, epsRel)
if err != nil {
return nil, err
}
omegaFit, err := tempCrit.OmegaFit(env, tempCrit.OmegaPlus)
if err != nil {
return nil, err
}
env.A, env.B = omegaFit[0], omegaFit[2]
env.PairCoeffsReady = true
// uncache env
env.D1, env.Mu_h, env.Mu_b, env.Beta = D1, Mu_h, Mu_b, Beta
}
// our guess for beta should be a bit above Beta_p
pairSystem, pairStart := tempPair.PairTempSystem(env)
_, err := solve.MultiDim(pairSystem, pairStart, epsAbs, epsRel)
if err != nil {
return nil, err
}
env.Beta += 0.1
// solve fluc temp system for reasonable values of Mu_h and D1 first
system, start := FlucTempD1MuSystem(env)
_, err = solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
// solve the full fluc temp system
system, start = FlucTempFullSystem(env)
solution, err := solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
return solution, nil
}