The article discusses the application of Green's functions in statistical physics, particularly focusing on two-time Green's functions. It begins by highlighting the increasing overlap between quantum field theory and statistical mechanics, where both fields share common challenges and methodologies. The text emphasizes the importance of Green's functions in studying interacting quantum fields and their utility in statistical mechanics, especially for large systems with a high number of particles. Key points include: 1. **Introduction to Two-Time Green's Functions**: The article introduces two-time Green's functions, which are extensions of correlation functions and are useful for analyzing systems with many interacting particles. These functions are defined through averages over the canonical ensemble of a large system. 2. **Equations for Green's Functions**: The article derives the equations governing two-time Green's functions, which are crucial for understanding the dynamics of interacting systems. These equations are derived by differentiating the averages of operator products with respect to time. 3. **Applications in Statistical Mechanics**: - **Non-equilibrium Processes**: The article discusses how two-time Green's functions can be used to study non-equilibrium processes, such as the reaction of a quantum system to an external perturbation. It provides a formula for the change in the average value of a dynamical variable due to the perturbation. - **Tensor Conductivity**: An example is given to illustrate the connection between the tensor conductivity and Green's functions. The conductivity is expressed in terms of the Fourier components of the retarded Green's function, which describes the influence of a periodic perturbation on the system. 4. **Ideal Quantum Gases**: The article concludes with a simple example of calculating Green's functions for ideal Fermi or Bose gases. It demonstrates how these functions can be derived directly from the Hamiltonian and the properties of the gases, without the need for complex averaging processes. Overall, the article emphasizes the importance of Green's functions in simplifying the analysis of complex systems in statistical physics, particularly in the context of non-equilibrium dynamics and the properties of ideal quantum gases.The article discusses the application of Green's functions in statistical physics, particularly focusing on two-time Green's functions. It begins by highlighting the increasing overlap between quantum field theory and statistical mechanics, where both fields share common challenges and methodologies. The text emphasizes the importance of Green's functions in studying interacting quantum fields and their utility in statistical mechanics, especially for large systems with a high number of particles. Key points include: 1. **Introduction to Two-Time Green's Functions**: The article introduces two-time Green's functions, which are extensions of correlation functions and are useful for analyzing systems with many interacting particles. These functions are defined through averages over the canonical ensemble of a large system. 2. **Equations for Green's Functions**: The article derives the equations governing two-time Green's functions, which are crucial for understanding the dynamics of interacting systems. These equations are derived by differentiating the averages of operator products with respect to time. 3. **Applications in Statistical Mechanics**: - **Non-equilibrium Processes**: The article discusses how two-time Green's functions can be used to study non-equilibrium processes, such as the reaction of a quantum system to an external perturbation. It provides a formula for the change in the average value of a dynamical variable due to the perturbation. - **Tensor Conductivity**: An example is given to illustrate the connection between the tensor conductivity and Green's functions. The conductivity is expressed in terms of the Fourier components of the retarded Green's function, which describes the influence of a periodic perturbation on the system. 4. **Ideal Quantum Gases**: The article concludes with a simple example of calculating Green's functions for ideal Fermi or Bose gases. It demonstrates how these functions can be derived directly from the Hamiltonian and the properties of the gases, without the need for complex averaging processes. Overall, the article emphasizes the importance of Green's functions in simplifying the analysis of complex systems in statistical physics, particularly in the context of non-equilibrium dynamics and the properties of ideal quantum gases.