The Nosé thermostat is explained in Chapter 6 of Understanding Molecular Simulation. Perhaps it's because I find it difficult to follow the derivation, but the physical meaning behind the $s$ parameter being added to the system isn't obvious to me.

The Langrangian of the system is extended and becomes:

$$\mathcal{L} = \sum^N_{i=1}\frac{m_i}{2}s^2\dot{\vec{r}}_i^2 - \mathcal{U}\left(\vec{r}^N\right) +\frac{Q}{2}\dot{s}^2 - \frac{L}{\beta}\ln(s)\tag{1}$$

Where $s$ is an additional coordinate, $Q$ is an effective mass associated with $s$, $L$ is a parameter to be fixed later, and presumably $\beta = \frac{1}{k_B T}$.

I understand that $s$ is supposed to represent the system being attached to a heat bath, but what is the physical explanation for this? How would you describe what the Nosé thermostat does qualitatively?

  1. Frenkel, D., & Smit, B. (2002). Understanding molecular simulation: From algorithms to applications. Academic Press. ISBN: 9780122673511
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    $\begingroup$ This is before all a mathematical trick, and I am not aware of a strict physical interpretation for it. There might be none (especially since it leads to non-Hamiltonian physics, makes me think it might just be a "weird trick"). I personally think of it as an extra fictitious particle that carries all the heat of the heat bath and has the magical ability to scale the speeds of the other particles with it. This is however just an image and not a physical description, of course $\endgroup$ Commented Sep 30, 2021 at 7:13
  • $\begingroup$ Okay thanks a lot, that is actually quite helpful! But it's a magical particle that is constantly interacting with all other particles in that case? Also, what does L represent to you? I understand Q is the measure of coupling? $\endgroup$
    – Connor
    Commented Sep 30, 2021 at 11:17
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    $\begingroup$ Q is the mass of the extra degree of freedom s (what you called the "magical particle"). $\endgroup$ Commented Oct 9, 2021 at 22:20
  • $\begingroup$ Do you mean the Nose thermostat specifically, or thermostats in general? $\endgroup$ Commented Oct 9, 2021 at 22:21
  • $\begingroup$ @PhilHasnip I mean specifically any thermostat that follows the Extended Lagrangian method. My own belief is that Nosé was aware of the Berendsen thermostat and simply changed the form of the frictional constant, then once that worked derived the Lagrangian, which is why it's hard to give it physical meaning! But if I'm wrong on the process, or if it can be given physical meaning, I would love to hear it! $\endgroup$
    – Connor
    Commented Oct 11, 2021 at 8:41