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.

Due to phenomenal progress in trapping and cooling of cold atoms, ultracold atoms has been a successful platform for studying fundamental physics and simulating quantum many-body systems with the advantage of tuning the parameters in real space. In this dissertation, we theoretically investigated inhomogeneous structures of different ultracold atomic mixtures in quasi-one-dimensional box potentials to explore the rich phases of atomic mixtures. The realization of uniform box potentials in experiments allows us to study inhomogeneous structures, phase diagrams and interface structures in various atomic mixtures which arise from the competition between interaction energy and kinetic energy. The presence of real space constraints like box potentials or quenching of interactions permits the extraction of characteristic length scale of the systems. We examined the characteristic lengths of spatial change such as, the healing lengths in the case of boson-fermion mixtures and correlation lengths in the case of superfluid-normal phase transition. The analyses of the healing lengths at the boson-fermion interface confirms the energy competition mechanism behind the phase-separation structures. We modelled spatially tunable inhomogeneous interactions for attractive Fermi gases to study analogs of the proximity effect and the spatial Kibble-Zurek mechanism (KZM) in a unified framework. The introduction of inhomogeneity lead to the distortion of the order parameter which is characterized by the correlation length. We extracted the correlation lengths from the pair wavefunction and correlation function to determine the penetration of the order parameter in the region where interaction vanishes. In the setup resembling the proximity effect, we found that the correlation lengths follow the BCS coherence length while only the exponent from the pair correlation function agrees to the Kibble-Zurek scaling in the case of spatial KZM setup. We also extract the critical exponent by adding a uniform bosonic background to the attractive Fermi gases and found that it does not alter the scaling behavior in the miscible phase. With recent experimental progress in the field of atomic mixtures and box potentials, our results will characterize the density profiles, correlation lengths and scaling behavior of those multi-component quantum systems.

I set up the path integral formalism to study interacting bosonic gases, binary boson-fermion mixtures, and fermionic superconductors. The large-N theory is applied to bosonic interactions. The corresponding auxiliary field theory is applied to fermionic interactions. Using the thermodynamic quantities obtained by evaluating the path integral, I map out the phase boundaries. For fermionic superconductors, the path integral method reproduces well-known methods that were developed previously.