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I have succesfully plotted the running of the gauge couplings to one loop order with mathematica.

My code is as follows:

SMbetafunctions = 
  {g1'[t] == 1/(16*Pi^2)*(41/6)*g1[t]^3,
   g2'[t] == 1/(16*Pi^2)*(-19/6)*g2[t]^3,
   g3'[t] == 1/(16*Pi^2)*-7*g3[t]^3,
   g1[0] == 0.35940,
   g2[0] == 0.64754,
   g3[0] == 1.1666};

sol = NDSolve[SMbetafunctions, {g1, g2, g3}, {t, 0, 50}];

Plot[Evaluate[{g1[t], g2[t], g3[t]} /. sol], {t, 0, 50},
  PlotLegends -> {g1, g2, g3},
  PlotRange -> Automatic ,
  PlotTheme -> "Automatic",
  Frame -> True,
  FrameLabel -> {"t = ln(μ/μ_0)", "Coupling Strength"}]]

The beta functions are defined as

$\qquad \frac{\partial g_1}{\partial t} = \frac{1}{16\pi}\frac{41}{6}g_1^3$

with $t = \log(\frac{\mu}{\mu_0})$.

So am I evaluating the ODE's over the variable $t$. But I want to express the x-axis in the energy $\mu$ instead.

How can I adjust the x-axis to reflect the energy?

I have tried to rewrite the RGE's in terms of $\mu$ instead of $t$ giving me

$\frac{\partial g_1}{\partial \mu} = \frac{1}{\mu} \frac{1}{16\pi}\frac{41}{6}g_1^3$

However, when I evaluate this, I get an $\frac{1}{0}$ error with, e.g., {mu, 1, 10000}.

With python for example I can easily adjust the values of the x-axis by $\mu = e^{t}\cdot\mu_0$ and which simply translates the values from to energy and reflects it also on the plot.

As an example, plotted with python what I have: enter image description here What I want to get: enter image description here

The values of mu indeed range from $[\approx 0, 1E22]$

Edit: Henrik's solution is what I was looking for:

LogLinearPlot[
 Evaluate[{g1[Log[μ]], g2[Log[μ]], g3[Log[μ]]} /. sol], {μ, 1, 10^22},
 PlotLegends -> {g1, g2, g3},
 Frame -> True,
 FrameLabel -> {"μ", "Coupling Strength"}
]

in my case $t = \ln(\frac{\mu}{\mu_0})$ instead of just $t = \ln(\mu)$, to achieve this one can just adjust

{g1[Log[μ]], g2[Log[μ]], g3[Log[μ]]}

to

{g1[Log[μ/μ_0]], g2[Log[μ/μ_0]], g3[Log[μ/μ_0]]}.

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closed as unclear what you're asking by m_goldberg, MarcoB, Coolwater, Sektor, Artes Feb 1 '18 at 18:04

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 4
    $\begingroup$ Probably ypu are looking for LogLinearPlot. LogPlot and LogLogPlot may be also of interest. $\endgroup$ – Henrik Schumacher Jan 27 '18 at 23:35
  • $\begingroup$ Thanks for your help, the plot should indeed be adjusted to one of those plots. However $t$ is evaluated in the range 0 to 50, while $mu$ should show the range 0 to $e^{50} \mu_0$, which is of the order ~1E22. $\endgroup$ – Reverie Jan 28 '18 at 0:03
  • $\begingroup$ The relation between t and μ is not clear in your post. Please edit the question to give a Mathematica expression for this relation. $\endgroup$ – m_goldberg Jan 28 '18 at 1:36
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A proposal without messing with the tickmarks: Just plot the functions you meant, namely $g_1 \circ \ln$, $g_2 \circ \ln$, and $g_3 \circ \ln$ (up to some constants) with a LogLinearPlot:

LogLinearPlot[
 Evaluate[{g1[Log[μ]], g2[Log[μ]], g3[Log[μ]]} /. sol], {μ, 1, 10^22},
 PlotLegends -> {g1, g2, g3},
 Frame -> True,
 FrameLabel -> {"μ", "Coupling Strength"}
]

enter image description here

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  • $\begingroup$ Dear Henrik, this is exactly what I was looking for! Thank you so much! I did not know it was possible to evaluate to Log as the only thing that came to mind was setting g1'[t] == 1/(16*Pi^2)*(41/6)*g1[t]^3 to g1'[Log[mu]] == 1/(16*Pi^2)*(41/6)*g1[Log[mu]]^3 Which when I evaluate it says I don't have a correct variable, but I did not consider your alternative. $\endgroup$ – Reverie Jan 28 '18 at 11:55
  • $\begingroup$ Glad to be of help! $\endgroup$ – Henrik Schumacher Jan 28 '18 at 11:56
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You could use Charting`ScaledTicks and Charting`ScaledFrameTicks:

Plot[
    Evaluate[{g1[t],g2[t],g3[t]}/.sol],
    {t,0,50},
    PlotLegends->{g1,g2,g3},
    Frame->True,
    FrameLabel->{"\[Mu]","Coupling Strength"},
    FrameTicks->{
        {Automatic,Automatic},
        {Charting`ScaledTicks[{Log,Exp}],Charting`ScaledFrameTicks[{Log,Exp}]}
    }
]

enter image description here

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xticks = {N @ Log[10^#], Superscript[10, #]}&/@Range[1, 22, 3];

Show[Plot[Evaluate[{g1[t], g2[t], g3[t]} /. sol], {t, 0, 50}], 
 Frame -> True, FrameTicks -> {{Automatic, Automatic}, {xticks, Automatic}}]

enter image description here

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