In this thesis, we study geometry-based non-equilibrium steady-state transport phenomena theoretically with the overarching goal to understand how the multi-path geometry can affect transport in classical and quantum systems. We begin with an overview of the physics associated with classical and quantum transport and the formalisms used to obtain results. In the non-equilibrium steady-state, one would expect that the local gradient imposed by the reservoirs would define a unique direction of flow from high-to-low. However, through this thesis we show that may not always be the case as one can devise a local steady-state atypical flow which goes from (low-to-high) by using a system with multi-path geometry. We address the universality of these steady-state local atypical flows in systems with multiple paths, through the following undertakings:

We show a classical harmonic system of Hookean springs and point masses coupled in a multi-path geometry driven by two Langevin reservoirs at different temperatures can give rise to a steady-state local atypical thermal flow. Through molecular dynamics simulations of Langevin equations for this system, we show that the atypical current depends on both internal and external parameters such as ratio of spring constants, ratio of masses and system-reservoir coupling respectively. We also show the robust nature of this atypical current against substrate induced non-linearity and asymmetric system-reservoir coupling. Two different approaches, namely the Redfield and Lindblad master equation, are used to extract the non-equilibrium steady-state thermal transport of a quantum system of oscillators coupled in a triangular geometry described in the coordinate-momentum space and as a Bose-Hubbard Hamiltonian respectively. Through the third quantization formalism and numerical simulation of the quantum master equations we show that atypical flows are universal to multi-path geometry and arise in both descriptions. We show that these atypical flows give rise to two patterns of internal steady-state circulations, clockwise and counterclockwise. We map out phase diagrams for these flow patterns as a function of system parameters thereby showing its robust nature. Finally, we show that these atypical flows and internal steady-state circulations are not limited to thermal transport but can be achieved for particle transport as well. We phenomenologically describe a hybrid system comprising of photonic structures and electronic quantum dots and show that the triangular geometry of this system can give rise to steady-state photonic circulations. We show the robust nature of these circulations against photon blockade and interactions through numerically calculated phase maps with the ratio of tunneling coefficients and system-reservoir coupling as the parameters.

At the end, we elaborate on the applications of these geometry-based steady-state atypical flows and outline possible experimental realizations to observe these atypical flows and circulations.