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I'm following an introduction to statistical mechanics and have seen the following definitions for fundamental temperature, pressure and chemical potential respectively:

$$\frac{1}{\tau} := \left( \frac{\delta \sigma}{\delta U} \right)_{V,N}$$ $$p := \tau \left( \frac{\delta \sigma}{\delta V} \right)_{U,N}$$ $$\mu := -\tau \left( \frac{\delta \sigma}{\delta N} \right)_{U,V}$$

where $\sigma$ is entropy $U$ energy, $V$ volume and $N$ the number of particles.

First, I suppose the inverse in the temperature definition is there because we expect lower temperatures to draw energy. But I can't explain the factor $\tau$ in the definitions of pressure and chemical potential, nor the minus sign in the latter. This is the introduction I read:

Consider two systems A and B with entropies $\sigma_A$ and $\sigma_B$ and volumes $V_A$ and $V_B$. They are put into contact so that they can exchange volume, but not energy or particles. The maximal entropy will then be obtained at:

$$ 0 = \frac{d\sigma_{AB}}{dV_A} = \frac{d\sigma_A}{dV_A} - \frac{d\sigma_B}{dV_B} \Leftrightarrow \frac{d\sigma_A}{dV_A} = \frac{d\sigma_B}{dV_B} $$

We call this equilibrium quantity $p/\tau$. But why not just $p$?

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This is thermodynamics in entropic formulation. The simple definitions (and more intuitive) of $\tau$,$p$ and $\mu$ are given in the energetic formulation. The entropic formulation is derived from the energetic one.

Think about $p/\tau$ as one function not as two separated entities.

In the energetic formulation U (internal energy of a system in equilibrium) is a function of $\sigma$,$V$ and $N$ and we define: $$\tau:=\frac{\partial U}{\partial \sigma}$$ $$p:=-\frac{\partial U}{\partial V}$$ $$\mu:=\frac{\partial U}{\partial N}$$

Equivalent to: $$dU=\tau d\sigma-pdV+\mu dN$$

With unjustified algebraic manipulations we obtain:

$$d\sigma=\frac{1}{\tau}dU+\frac{p}{\tau}dV-\frac{\mu}{\tau} dN$$

hence $$\frac{\partial \sigma}{\partial U}=\frac{1}{\tau}$$ $$\frac{\partial \sigma}{\partial V}=\frac{p}{\tau}$$ $$\frac{\partial \sigma}{\partial N}=-\frac{\mu}{\tau}$$

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