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Analysis of the semiclassical (SC) momentum rate equations
Plotting the ICs and BCs and examining sensitivity to the Re and Im forces
MRB: 24/2/2020, 27/2/2020, 2/3/2020.
We examine solution of the SC version of the momentum rate equations, in which O terms for  are removed. A high level of sensitivity to ICs and BCs makes solution finding difficult.
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with(PDETools): with(CodeTools):with(plots):
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We set up the initial conditions:
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ICu := {u(x, 0) = .1*sin(2*Pi*x)}; ICv := {v(x, 0) = .2*sin(Pi*x)};
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(1) |
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plot([0.1*sin(2*Pi*x),0.2*sin(Pi*x)],x = 0..2, title="ICs:\n u(x,0) (red), v(x,0) (blue)",color=[red,blue],gridlines=true);
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The above initial conditions represent a positive velocity field  (blue) and a colliding momentum field  (red).
Here are the BCs
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BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t))};
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(2) |
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BCv := {v(0,t) = 0.5*sin(2*Pi*t),v(2,t)=-0.5*sin(2*Pi*t)};
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(3) |
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plot([0.5*(1-cos(2*Pi*t)),0.5*sin(2*Pi*t),-0.5*sin(2*Pi*t)],t=0..1,color=[red,blue,blue],linestyle=[dash,dash,dot],title="BCs:\n u(0,t) (red-dash),\n v(0,t) (blue-dash), v(1,t) (blue-dot)",gridlines=true);
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We can now set up the PDEs for the semiclassical case.
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hBar:= 1:m:= 1:Fu:= 0.2:Fv:= 0.1:#1.0,0.2
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pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;
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(4) |
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pdev := diff(v(x,t),t)+u(x,t)/m*(diff(v(x,t),x))-hBar*(diff(u(x,t),x$2))/(2*m)+v(x,t)*(diff(u(x,t),x))/m = Fv;
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(5) |
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ICu:={u(x,0) = 0.1*sin(2*Pi*x)};
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(6) |
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ICv:={v(x,0) = 0.2*sin(Pi*x/2)};
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(7) |
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(8) |
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BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t)), D[1](u)(2,t) = 0.1*cos(2*Pi*t)};
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)(2, t) = .1*cos(2*Pi*t)}](/view.aspx?sf=228990_question/86eef5cedc4ca9bc0c8a0681bcb758ed.gif)
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(9) |
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BCv := {v(0,t) = 0.2*(1-cos(2*Pi*t))};
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(10) |
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)(2, t) = .1*cos(2*Pi*t)}](/view.aspx?sf=228990_question/dc109fd9eb2068611bbf640f68c5aa96.gif)
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We now set up the PDE solver:
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pds := pdsolve({pdeu,pdev},{BC[],IC[]},time = t,range = 0..2,numeric);#'numeric' solution
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(12) |
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Cp:=pds:-animate({[u, color = red, linestyle = dash],[v,color = blue,linestyle = dash]},t = 30,frames = 400,numpoints = 400,title="Semiclassical momentum equations solution for Re and Im momenta u(x,t) (red) and v(x,t) (blue) \n under respective constant positive forces [0.2, 0.1] \n with sinusoidal boundary conditions at x = 0, 1 and sinusoidal initial conditions: \n time = %f ", gridlines = true,linestyle=solid):Cp;
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(13) |
Observations on the quantum case:
The classical equation for  is independent of the equation for  .  (red) is a solution of the classical Burgers equation subject to a force 0.2, but  is NOT influenced by  . On the otherhand,  (blue) is a solution of the quantum dynamics equation subject to force 0.1 and is influenced by  . This one way causality (u  ) is a feature of the semiclassical case, and it emphasises the controlling influence of the classical  , which modulates the quantum solution for  . Causally, we have u .
The initial conditions are of low momentum amplitude: 0.1 for the classical  (red) field and .2 for  (blue) but their influence is soon washed out by the boundary conditions  and  that drive the momentum dynamics.
The temporal frequency of the boundary condition on the  -field is twice that of the classical  -field. This is evident in the above blue transient plot. Moreover, the boundary condition on the classical  -momentum (red), drives that field in the positive direction, initially overtaking the quantum  field, as consistent with the applied forces [0.2, 0.1].  Although initially of greater amplitude than the classical  field, the  momentum field is asymptotically of the same amplitude as the  field, but has greater spatial and temporal frequency, owing to the boundary conditions.
Referring to the semiclassical momentum rate equations, we note that the classical field  (red) modulates the quantum momentum rate equation for  .
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