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Simulation of Crystalline Nanoporous Materials and the Computation of Adsorption/Diffusion Properties
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
Isotherms can be computed in the GE. One system represents the adsorbed phase and contains a nanoporous framework (either rigid or flexible) with positions obtained from experiments or QM, and the other box represents the bulk-fluid phase. Alternatively, if the isotherm is computed as a function of fugacity (instead of pressure), or an accurate equation of state (EOS) is known, then the computation of the fluid phase can be avoided and replaced by an imposed fugacity f. Therefore, the most common ensemble for adsorption computations is the grand-canonical ensemble where the volume V is fixed, and the temperature T and the chemical potential μ (directly related to the fugacity via μ = μ0 + RT ln[ f/f0]) are imposed. The grand-canonical ensemble describes a system in contact with a heat and particle bath. Note that the chemical potential has energy units, while fugacity has pressure units. Pressure p is related to fugacity f by f = ϕp, where ϕ is the fugacity coefficient (computed, e.g., from an EOS). In the low pressure regime, pressure and fugacity are equal.
Z-Partition Function
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles (Nnrepresents total particle number) with the environment, at a fixed temperature, volume, and chemical potential μ: ()Z=∑n=0n=Ne−β(En−Nnμ).
NEGF Method for Design and Simulation Analysis of Nanoscale MOS Devices
Published in Ashish Raman, Deep Shekhar, Naveen Kumar, Sub-Micron Semiconductor Devices, 2022
The second quantization describes the many-body systems by quantizing the fields using a basis that describes the number of particles occupying each state. The many-body systems are described by the second quantization scheme, in which fields are quantized after assuming that each state is occupied and described by several particles [35]. Using this quantization concept, a one-body state is articulated in terms of occupation number. For any order, a complete one-body basis is given as vi and for an N-body system, the basis states can be represented as nvi,nv2,... in the occupation number representation. This notation means nvi particles in the state vi with ∑invi=N [29, 33]. To draw up statistical mechanics in the form of the grand-canonical ensemble (a thermodynamic ensemble that describes a system in contact with a heat and particle bath, where chemical potential and the temperature are defined while the particle number and energy of the particles are not specified), one should treat each state with a distinct number of particles. Therefore, Fock space F is used to define the direct addition of all N-body Hilbert spaces [36]. f=H0⊕H1⊕H2.....
Correlation functions in mixtures with energetically favoured nearest neighbours of different kind: a size-asymmetric case
Published in Molecular Physics, 2021
O. Patsahan, A. Meyra, A. Ciach
The objective of our study in this work is twofold. First, we want to determine how the size asymmetry influences the periodic arrangement of particles, when the neighbourhood of different particles is energetically favoured. Our second goal is to verify the accuracy of the modified DFT for different size ratios of the particles, and for different thermodynamic states. The theoretical results are compared with Monte Carlo (MC) simulations. The simulations are performed in the canonical ensemble, while in the theory, the grand-canonical ensemble is used. Even though fluctuations of the concentration in mesoscopic regions play a significant role in our system, fluctuations of the numbers of the big and small particles in the macroscopic volume are negligible far from the gas–liquid separation. For this reason, the canonical and the grand-canonical ensembles are expected to give equivalent results for the structure of the liquid phase.
Order-preserving dynamics in one dimension – single-file diffusion and caging from the perspective of dynamical density functional theory
Published in Molecular Physics, 2021
René Wittmann, Hartmut Löwen, Joseph M. Brader
Having established an ensemble which provides ordered distributions with a fixed particle number, we ask the question of what is (are) the corresponding grand canonical ensemble(s), in which the particle numbers can fluctuate. While the canonical partition function of the ordered ensemble is equal to that of indistinguishable particles (only the ensemble averages of nontrivial operators may be different), it turns out that the corresponding grand partition function is neither unique (there are different sensible ways to introduce such a quantity) nor equivalent to the grand partition function of a single component with the chemical potential μ. To see this, we will first define below the two possible partition functions and corresponding to our ordered three-component mixture, assuming that the particle number fluctuates in two and three species, respectively (as indicated by the subscripts ‘’ for canonical and ‘’ for grand canonical treatment of a species). A third possibility, , with one fluctuating species could also be considered but does not turn out to be useful in the present context. Recall that up to this point the particle number of the species holding the tagged particle has been fixed as .