MODEL: Hedgehog Dynamics
Axial Diffusion of Orientation determined by gradient in Constant of Integration, a

 m :=sin(a);

[Maple Math]

 y := sqrt(1-m^2*sin(x)^2);

[Maple Math]



Compute Elliptic Integrals
    Second Kind
 E :=int(y,x = 0..Pi/2); plot(E,a=0..Pi/2, title = `Complete Elliptic Integral of the Second Kind`);

[Maple Math]

Write files: EllipticE
E_new := unapply(E,a): evalf(E_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 0.0 +(i-1)*0.0157: new[i,2] := evalf(E_new(0.0 +(i-1)*0.0157)): od:
new[1,2];
 fd := fopen(Eel.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]

    First Kind
 K :=int(1/y,x = 0..Pi/2); plot(K,a=0..Pi/2, title=`Complete Elliptic Integral of the First Kind`);

[Maple Math]

Write File: EllipticK
 K_new := unapply(K,a): evalf(K_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 0.0 +(i-1)*0.0157: new[i,2] := evalf(K_new(0.0 +(i-1)*0.0157)): od:
 new[1,2];
[Maple Math]
 fd := fopen(Kel.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]
 

Constant of Integration: 1/sin(a)^2

Total Energy(sin(a)) - No dynamics from surface terms.
 F := (1/sin(a))*(2*E - (cos(a)^2)*K) ; plot(F,a=1.3..1.85, title = `Free Energy/PiK - per unit length`);

Write file: Energy
F_new := unapply(F,a): evalf(F_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 1.3 +(i-1)*0.005: new[i,2] := evalf(F_new(1.3 +(i-1)*0.005)): od:
 new[1,2];
[Maple Math]
 fd := fopen(Energy.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]

Show Minimum: a = pi/2
 z := diff(F,a): plot(z,a=0.0..Pi/2, title = `dF/da`);

Write to file:
 z_new := unapply(z,a): evalf(z_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := .01+(i-1)*0.0156: new[i,2] := evalf(z_new(0.01 +(i-1)*0.0156)): od:
 new[1,2];
 fd := fopen(zbig.txt, WRITE); writedata(fd, new, float); fclose(fd);

[Maple Math]

Expanded Plot
 plot(z,a=1.4..1.7, title = `dF/da =0 `);

Write to File:
 z_new := unapply(z,a): evalf(z_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 1.40+(i-1)*0.003: new[i,2] := evalf(z_new(1.4+(i-1)*0.003)): od:
 new[1,2];
[Maple Math]
 fd := fopen(z.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]
 
 

Evaluate distance from center, ro, where n is axial

r=R, n is radial; r=ro n is axial
ro = 0 when a = /2
r0 := exp(-sin(a)*EllipticK(csgn(sin(a))*sin(a))(a)); plot(r0, a=0..Pi/2, title= `r_o/R vs. a`);

[Maple Math]

Write file:
 r0_new := unapply(r0,a): evalf(r0_new(1.57));
[Maple Math]
 new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 0+(i-1)*0.015: new[i,2] := evalf(r0_new(0+(i-1)*0.015)): od:
 new[1,2];
[Maple Math]
 fd := fopen(r0.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]
 

MODEL

Axial diffusion constant, Dz, determined by gradient in a
Relevant length scale is ro

 Dz := -z/(r0)^2: plot(vz,a=0..Pi/2, title=`v_z vs. a`);

Dz  accelerates when a -> pi/2
Dz nearly constant when a < 1.5

 plot(Dz,a=1.2..Pi/2, title=`v_z vs. a: r_o small`);

Write files
 Dz_new := unapply(Dz,a): evalf(Dz_new(1.57));

[Maple Math]
new := array(1..101,1..2): for i from 1 to 101 do new[i,1] := 1.2+(i-1)*0.0037: new[i,2] := evalf(vz_new(1.2+(i-1)*0.0037)): od:
 new[1,2];
[Maple Math]
 fd := fopen(Dz.txt, WRITE); writedata(fd, new, float); fclose(fd);
[Maple Math]
Compute acceleration
 new2 := array(1..100,1..2): for i from 1 to 100 do new2[i,1] := 1.2+(i-1)*0.0037: new2[i,2] := (new[i+1,2]-new[i,2])/0.0037: od:
 fd := fopen(vz_diff.txt, WRITE); writedata(fd, new2, float); fclose(fd);
[Maple Math]

 

Ref: P. E. Cladis and M. Kléman, Non-singular Disclinations of Strength S=+1 in Nematics, J. de Physique 33, 591 (1972).