Zeros of the partition function

[mhill]

given a partition function

$$Tr[e^{-BH}]$$ or $$Z(B)= \int_{P}dx dp e^{-BH(p,q)}$$

is there any meaning for its zeros ? , i mean what happens in case the partition function Z(B)=0 for some 'B' or temperature $$B=1/kT$$ do these zeros have a meaning ?? thanks

 

[olgranpappy]

what you have written is positive definite so it has no zeros (i.e., is never zero)... assuming beta is real... and assuming H is a true Hamiltonian...

 

[denian]

The zeros of the partition function have significant implications in statistical mechanics and thermodynamics. In particular, the zeros of the partition function can provide information about the phase transitions of a system.

When the partition function becomes zero at a certain value of temperature or energy, it signifies a phase transition in the system. This means that the system undergoes a sudden change in its properties, such as its energy, entropy, or order. The temperature or energy value at which the partition function becomes zero is known as the critical point, and it marks the boundary between two different phases of the system.

For example, in the Ising model of magnetism, the partition function becomes zero at a certain temperature known as the Curie temperature. This is the temperature at which the system undergoes a phase transition from a ferromagnetic phase to a paramagnetic phase. Similarly, in the Van der Waals equation of state, the partition function becomes zero at the critical temperature, which marks the transition from a gas phase to a liquid phase.

Furthermore, the study of the zeros of the partition function can also provide insights into the behavior of the system near the critical point. For instance, the distribution of the zeros can reveal the universality class of the system, which is a fundamental concept in critical phenomena.

In short, the zeros of the partition function are of great importance in understanding the behavior and phase transitions of a system. They provide valuable information about the critical points and universality classes of the system, which in turn can help us gain a deeper understanding of its properties and behavior.