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Entropy Increase in Irreversible Processes

In any real, irreversible process, the total entropy of the system and its surroundings always increases.
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The statement of the theorem

Let S\mathcal{S} be the system and R\mathcal{R} be the surroundings. Consider a process P\mathcal{P} occurring between initial state X1\mathbf{X}_1 and final state X2\mathbf{X}_2. The total entropy change is defined as ΔStotal=ΔSS+ΔSR\Delta S_{total} = \Delta S_{\mathcal{S}} + \Delta S_{\mathcal{R}}. For any irreversible process P\mathcal{P}, the entropy production S˙gen\dot{S}_{gen} is defined by the rate of change of total entropy: S˙gen=dStotaldt\dot{S}_{gen} = \frac{d S_{total}}{d t}. The Second Law of Thermodynamics dictates that the entropy production must be non-negative, S˙gen≥0\dot{S}_{gen} \ge 0. For a strictly irreversible process, the inequality is strict: ΔStotal=∫t1t2S˙gendt>0\Delta S_{total} = \int_{t_1}^{t_2} \dot{S}_{gen} dt > 0. Specifically, if the process involves irreversible fluxes Ji\mathbf{J}_i driven by generalized forces Xi\mathbf{X}_i (e.g., heat flux Jq\mathbf{J}_q driven by temperature gradient ∇T\nabla T), the entropy production rate is given by the sum of products of fluxes and forces: S˙gen=∑iJi⋅Xi>0\dot{S}_{gen} = \sum_{i} \mathbf{J}_i \cdot \mathbf{X}_i > 0