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]]
]
Meq
andM0
? $\endgroup$ – zhk Mar 14 '19 at 14:42