How to plot a single plot with two variables

Question

guys.

Suppose I have this equation:

M(t) = Meq + (M0 - Meq) exp[-t/τ]

where t denotes de time, Meq and M0 defined parameters and τ is an another defined parameter.

I can easily plot this equation (as a function of time) defining fixed values for Meq, M0 and τ, and this results in the plot shown below:

But my real problem is: I need to plot the same equation (also as a funtion of time), but not for only one fixed value of τ, but actually to a distribution (a Log-Normal Distribution) of values of τ.

The result should be a plot with the same 'behaviour' of the plot for fixed values of τ. It should be something this way:

If someone could help me, I would appreciate very much.

Thanks in advance :)

Answer 1

If I understand correctly, you want to check the magnetization plots for different distributions of $\tau$.

This would involve first generating the distribution function that obtains a certain distribution for $\tau$, let's start with the NormalDistribution.

dist[x_] := PDF[NormalDistribution[1, 0.4], Log@x];
taulist = Table[dist[x], {x, -6, 60, 4}][[3 ;;]];
Dimensions@taulist
N@taulist

(* Output

{15}

{0.743123, 0.140625, 0.00496708, 0.00022532, 0.0000140927, 
 1.16059*10^-6, 1.1985*10^-7, 1.49193*10^-8, 2.17127*10^-9, 
 3.60749*10^-10, 6.71557*10^-11, 1.37989*10^-11, 3.0918*10^-12, 
 7.47922*10^-13, 1.93718*10^-13}

*)

Or a LogNormalDistribution which you are actually interested in such as:

distlognorm[x_] := PDF[LogNormalDistribution[-1, 0.005], Log@x];
taulistlognorm = 
  Table[distlognorm[x], {x, 1.43, 1.46, 0.0020}][[2 ;;]];
Dimensions@taulistlognorm
N@taulistlognorm

(* Output

{15}

{0.0017637, 0.0558546, 0.955951, 8.92674, 45.9087, 131.231, 210.398, \
190.881, 98.8533, 29.4746, 5.1026, 0.517151, 0.0309356, 0.00110101, \
0.0000234976}

*)

Now we can use these values to define $\tau$ and the main function that you have suggested in the question.

Since you don't specify, I chose $M_{eq} = 1$ and $M0 = -1$, but you can of course change these.

Mt[t_, \[Tau]_] := Module[{Meq = 1, M0 = -1},
   Return[Meq + (M0 - Meq) Exp[-t/\[Tau]]];
   ];

Using this function Mt[x], we can plot the two distributions and check it's dependence on the variation with $\tau$.

Code for the plots is given below:

Module[{},
 pltlist = {};
 Do[
  AppendTo[pltlist, 
   LogLinearPlot[{Mt[t, taulist[[ntau]]]}, {t, Exp[-100], Exp[100]}, 
    PlotStyle -> ColorData[ntau, "ColorList"]]]
  , {ntau, 1, 15}];
 pltnorm = Show[pltlist, Frame -> True, FrameLabel -> {"t,(sec)", ""}, 
 FrameStyle -> Directive[Black, FontSize -> 16]]
 ]

Module[{},
 pltlist = {};
 Do[
  AppendTo[pltlist, 
   LogLinearPlot[{Mt[t, taulistlognorm[[ntau]]]}, {t, Exp[-100], Exp[100]}, 
    PlotStyle -> ColorData[ntau, "ColorList"]]]
  , {ntau, 1, 15}];
 pltlognorm = Show[pltlist, Frame -> True, FrameLabel -> {"t,(sec)", ""}, 
 FrameStyle -> Directive[Black, FontSize -> 16]]
 ]

Answer 2

I have an idea for your plot

Meq = 0.5; 
M0 = 1; 
Plot[Evaluate[{Meq + (M0 - Meq)/E^(t/tau)} /. tau -> {2, 4, 6}], {t, -10, 4}, PlotRange -> All]

Thereby you can specify different values for tau and plot them together. By the way the function you are sharing with us is an exponential decay function, and not a sigmoidal or logistic function. I hope the pictures you shared are only examples and you don't expect a sigmoidal result?!

A sigmoidal function would look like:

1/(1+e^-t)

I don't understand what you want to do with the LogNormalDistribution.