Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics

The elsewhere surmized topological origin of phase transitions is given here important evidence through the analytic study of an exactly solvable model for which both topology of submanifolds of configuration space and thermodynamics are worked out. The model is a mean-field one with a $k$-body interaction. It undergoes a second-order phase transition for $k=2$ and a first-order one for $k>2$. This opens a perspective for the understanding of the deep origin of first and second-order phase transitions, respectively. In particular, a remarkable theoretical result consists of a mathematical characterization of first-order transitions. Moreover, we show that a “reduced” configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical $N$-body system and its macroscopic thermodynamic behavior. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic $2N$-dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

Figure 1

Temperature $T$ as a function of canonical average energy $e$ for three different values of $k$; for $k=1$ there is no phase transition, while for $k=2$ there is a second order transition and for $k>2$ a first order one.

Figure 2

Canonical average potential energy $v$ as a function of canonical average energy $e$ for $k=1$, 2 and 3. The upper phase transition point is, for $\forall k⩾2$, $v_c=\Delta$.

Figure 3

Temperature $T$ as a function of canonical average potential energy $v$ for three different values of $k$.

Figure 4

Microcanonical temperature $T$ as a function of energy $e$ for three different values of $k$; for $k=1$ there is no phase transition, while for $k=2$ there is a second order transition and for $k>2$ a first order one.

Figure 5

Microcanonical average potential energy $v$ as a function of energy $e$ for $k=1,2$, and 3. The phase transition point is, for $\forall k⩾2$, $v_c=\Delta$.

Figure 6

Microcanonical temperature $T$ as a function of microcanonical average potential energy energy $v$ for three different values of $k$.

Figure 7

Logarithmic Euler characteristic of the $M_v$ manifolds $\sigma \left(v\right)$ (see text) as a function of the potential energy $v$. The phase transition is signaled as a singularity of the first derivative at $v_c=\Delta$; the sign of the second derivative around the singular point allows to discriminate between transitions of different order. The region $v>\Delta$, in which ${\sigma }^{\prime }\left(v\right)<0$, in not reached by the system (see text).

Figure 8

The submanifolds ${\mathcal{M}}_v$ in the case $k=1$ for $v=0.5\Delta ,\Delta ,1.5\Delta ,2\Delta$ (from left to right). All the submanifolds are topologically equivalent to a single disk.

Figure 9

The submanifolds ${\mathcal{M}}_v$ in the case $k=2$ for $v=0.5\Delta ,\Delta ,1.5\Delta ,2\Delta$ (from left to right). As $v<v_c=\Delta$ the submanifolds are topologically equivalent to two disconnected disks, while as $v>v_c$ they are equivalent to a single disk.

Figure 10

The submanifolds ${\mathcal{M}}_v$ in the case $k=3$ for $v=0.5\Delta ,\Delta ,1.5\Delta ,2\Delta$ (from left to right). As $v<v_c=\Delta$ the submanifolds are topologically equivalent to three disconnected disks, while as $v>v_c$ they are equivalent to a single disk.

Figure 11

Logarithm of the sum of the Morse indexes divided by $N$, $\mu =\left(1∕N\right)\log \left[{\mu }_0+2{\Sigma }_{i=1}^{N-1}{\mu }_i\left(M_v\right)+{\mu }_N\right]$ of the manifolds $M_v$ vs the energy density $v$, scaled with $\Delta$, for $k=1,2,3$.