he same because the hot-mode gas cools down to the same temperature as the cold-mode gas.

Although it takes some time to reach pressure equilibrium if the hot gas is suddenly heated, we expect the hot and cold gas to be approximately in equilibrium inside the halo, because a phase with a higher pressure will expand, lowering its pressure, and compressing the phase with the lower pressure, until equilibrium is reached. While the pressure distributions do overlap, there is still a significant difference between the two. This difference decreases somewhat with increasing resolution, because the cold gas reaches higher densities and thus higher pressures, as is shown in the appendix. From the example pressure map (Fig. \[fig:halo\]) we can see that the filaments inside the halo are in fact approximately in pressure equilibrium with the diffuse gas around them. At first sight this seems at odds with the fact that the median pressure profiles are different. However, the pressure map also reveals an asymmetry in the pressure inside the halo, with the gas to the left of the centre having a higher pressure than the gas to the right of the centre. Because there is also more hot-mode gas to the left, this leads to a pressure difference between the two modes when averaged over spherical shells, even though the two phases are locally in equilibrium. The asymmetry arises because the hot-mode gas is a space-filling gas and the flow has to converge towards the centre of the halo, which increases its pressure. The cold-mode gas is not space filling and therefore does not need to compress as much.

Entropy
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We define the entropy as $$S\equiv \dfrac{P(\mu m_\mathrm{H})^{5/3}}{k_\mathrm{B}\rho^{5/3}},$$ where $\mu$ is the mean molecular weight, $m_\mathrm{H}$ is the mass of a hydrogen atom, and $k_\mathrm{B}$ is Boltzmann’s constant. Note that the entropy remains invariant for adiabatic processes. In the central panel of Fig. \[fig:halo\] we clearly see that the filaments have much lower entropies than the diffuse gas a