In my previous post, I computed the entropy generation rate during my constant-entropy r-process calculation. The calculation showed substantial entropy generation at certain points during the calculation. In this post, I will seek to understand that entropy generation.

I will begin by considering the entropy generation near *T _{9} = 5*, which is a local peak in the entropy generation. I note that I can infer that this entropy generation is not due to weak decays from the figure in the previous post. From the file

**cd nucnet-tools-code/examples/thermo**

**./examples_make**

Again, if you have trouble with clang, you can simply type

**make compute_sdot_by_reactions**

Once the code is compiled, I can compute the reactions that contribute to the entropy generation rate at step 30 by typing

**./compute_sdot_by_reaction ../../my_examples/network/my_output1.xml --zone_xpath "[position() = 30]"**

The zone XPath selects step number 30. The output from my calculation gives a long list headed by

time(s) = 0.0844315 t9 = 4.77324 rho(g/cc) = 135551 sdot = 8.10825 he4 + ar46 -> n + ca49 7.993122e-01 n + mg25 -> mg26 + gamma 5.583167e-01 he4 + fe62 -> n + ni65 3.390110e-01 he4 + ar44 -> n + ca47 3.368543e-01

From this, I see that the reaction * ^{4}He + ^{46}Ar → n + ^{49}Ca* is the leading contributor to the entropy generation rate. I will seek to understand this.

First, I note that the change in the entropy in a system is given by

Here *T* is the temperature, *ds* is the change in the entropy per nucleon, *dq* is the heat gained or lost by the system per nucleon, *μ _{i}* is the chemical potential of species

It is convenient to consider the chemical potential of the nuclear species in terms of an offset from the value in nuclear statistical equilibrium. I thus can write

The other species in the system are the electrons and positrons and neutrinos. The net electron-to-nucleon ratio is *Y _{e}* and is given by

where *Y _{e-}* is the abundance per nucleon of electrons and

where *ν _{e-}* refers to neutrinos associated with electrons (referred to as "electron-type neutrinos") and

I now consider the reaction * ^{4}He + ^{46}Ar → n + ^{49}Ca*. This reaction is not a weak reaction, so the net electron-to-nucleon ratio and the net electron-type neutrino abundance do not change. The rate of change of the abundance of

*N _{A}* is Avogadro's number,

I now compute the relevant terms. I change into the analysis directory and type

**./examples_make**

I then compute the chemical potential offsets from the NSE values for step 30 by typing

**cd ../analysis**

**./compute_zone_mu ../../my_examples/network/my_output1.xml "[position() = 30]" > mu_30.txt**

The output file *mu_30.txt* contains a header and then columns giving the species, *Z*, *A*, the abundance, and the chemical potential offset divided by *kT*. From the data, I find the *Δ* values for * ^{4}He, ^{46}Ar*, and

**./compute_flows ../../my_examples/network/my_output1.xml "[position() = 30]" > flows_30.txt**

I search the output file *flows_30.txt* for the reaction and find the line

he4 + ar46 -> n + ca49 5.921e-02 8.128e-08 5.921e-02

The forward flow is 5.921e-2. The reverse flow is 8.128e-8. The net flow is thus 5.921e-2. The entropy generation rate per nucleon from this reaction is therefore 13.498718 x 5.921e-2 = 0.7993, in agreement with our calculation above.

Entropy is being generated at this point in the calculation because the large equilibrium present is breaking down into smaller clusters. The species * ^{46}Ar* and

Anonymous