Quantum transport phenomena refer to the movement of particles, such as electrons, in systems where quantum effects play a significant role. These phenomena are particularly important in molecular and nanoscale systems, where the dimensions are comparable to the particles' de Broglie wavelengths. In these systems, quantum effects like wave-particle duality, quantum tunneling, and quantum interference become significant.Differences in quantum transport between molecular and nanoscale systems:1. Size and dimensionality: Molecular systems are typically smaller and have lower dimensionality than nanoscale systems. This leads to stronger quantum confinement effects in molecular systems, which can result in discrete energy levels and quantized conductance.2. Complexity: Molecular systems are often more complex than nanoscale systems, with a larger number of interacting particles and degrees of freedom. This can lead to more complex quantum transport behavior, such as many-body effects and electron-phonon interactions.3. Disorder and fluctuations: Molecular systems are often more disordered and subject to thermal fluctuations than nanoscale systems, which can lead to additional sources of quantum interference and localization effects.To quantitatively describe and predict quantum transport phenomena in molecular and nanoscale systems, various probabilistic and statistical models can be employed. Some of the commonly used models include:1. Non-equilibrium Green's function NEGF formalism: This is a powerful method for describing quantum transport in molecular and nanoscale systems. It involves calculating the Green's functions of the system, which describe the probability amplitudes for particle propagation, and using them to compute transport properties such as conductance and current.2. Landauer-Büttiker formalism: This is a simpler approach based on the scattering matrix, which relates the incoming and outgoing wave functions of particles in the system. The Landauer-Büttiker formalism can be used to calculate the conductance and other transport properties of molecular and nanoscale systems.3. Quantum Monte Carlo QMC methods: These are numerical techniques that use random sampling to solve the Schrödinger equation and compute the quantum transport properties of molecular and nanoscale systems. QMC methods can account for many-body effects and electron-phonon interactions, making them suitable for complex systems.4. Tight-binding models: These are simple models that describe the quantum transport properties of molecular and nanoscale systems using a small number of parameters, such as the hopping integrals between neighboring sites and the on-site energies. Tight-binding models can be used to study the effects of disorder and fluctuations on quantum transport.By employing these models and methods, researchers can quantitatively describe and predict the quantum transport phenomena in molecular and nanoscale systems, which can help in the design and optimization of novel devices and materials with tailored transport properties.