Sandipan Dutta

Classical representation of a quantum system at equilibrium: Theory

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities, and the shell structure of harmonically bound charges.

Uniform electron gas at warm, dense matter conditions

A simple, practical model for computing the equilibrium thermodynamics and structure of the uniform electron gas (jellium) by classical strong-coupling methods is proposed. Conditions addressed are those of interest for recent studies of warm dense matter: solid densities and temperatures from zero to plasma states. An effective pair potential and coupling constant are introduced, incorporating the ideal gas, low density, and weak-coupling quantum limits. The resulting parameter-free, analytic model is illustrated by the calculation of the pair correlation function via strong-coupling classical liquid state theory. The results compare favorably with the first finite-temperature restricted path integral Monte Carlo simulations reported recently.

Classical representation of a quantum system at equilibrium: Applications

In the preceding paper, the structure and thermodynamics of a given quantum system was represented by a corresponding classical system having an effective temperature, local chemical potential, and pair potential. Here, that formal correspondence is implemented approximately for applications to two quantum systems. The first is the electron gas (jellium) over a range of temperatures and densities. The second is an investigation of quantum effects on shell structure for charges confined by a harmonic potential.

Classical Representation of a Quantum System at Equilibrium

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Green function of correlated genes in a minimal mechanical model of protein evolution

Significance Many protein functions involve large-scale motion of their amino acids, while alignment of their sequences shows long-range correlations. This has motivated search for physical links between genetic and phenotypic collective behaviors. The major challenge is the complex nature of protein: nonrandom heteropolymers made of 20 species of amino acids that fold into a strongly coupled network. In light of this complexity, simplified models are useful. Our model describes protein in terms of the Green function, which directly links the gene to force propagation and collective dynamics in the protein. This allows for derivation of basic determinants of evolution, such as fitness landscape and epistasis, which are often hard to calculate. The function of proteins arises from cooperative interactions and rearrangements of their amino acids, which exhibit large-scale dynamical modes. Long-range correlations have also been revealed in protein sequences, and this has motivated the search for physical links between the observed genetic and dynamic cooperativity. We outline here a simplified theory of protein, which relates sequence correlations to physical interactions and to the emergence of mechanical function. Our protein is modeled as a strongly coupled amino acid network with interactions and motions that are captured by the mechanical propagator, the Green function. The propagator describes how the gene determines the connectivity of the amino acids and thereby, the transmission of forces. Mutations introduce localized perturbations to the propagator that scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. We find that epistasis—the interaction among mutations in the gene—is related to the nonlinearity of the Green function, which can be interpreted as a sum over multiple scattering paths. We apply this mechanical framework to simulations of protein evolution and observe long-range epistasis, which facilitates collective functional modes.

Quantum potential for diffraction and exchange effects

Semiclassical methods of statistical mechanics can incorporate essential quantum effects by using effective quantum potentials. An ideal Fermi gas interacting with an impurity is represented by a classical fluid with effective electron–electron and electron-impurity quantum potentials. The electron-impurity quantum potential is evaluated at weak coupling, leading to a generalization of the Kelbg potential to include both diffraction and degeneracy effects. The electron–electron quantum potential for exchange effects only is the same as that discussed earlier by others.

Strong-coupling electrostatic theory of polymer counterions close to planar charges.

Strong-coupling phenomena, such as like-charge macroion attraction, opposite-charged macroion repulsion, charge renormalization, and charge inversion, are known to be mediated by multivalent counterions. Most theories treat the counterions as point charges and describe the system by a single coupling parameter that measures the strength of the Coulomb interactions. In many biological systems, the counterions are highly charged and have finite sizes and can be well-described by polyelectrolytes. The shapes and orientations of these polymer counterions play a major role in the thermodynamics of these systems. In this work we apply a field-theoretic description in the strong-coupling regime to the polymer counterions in the presence of a fixed charge distribution. We work out the special cases of rodlike polymer counterions confined by one, and two charged walls, respectively. The effects of the geometry of the rodlike counterions and the excluded volume of the walls on the density, pressure, and free energy of the rodlike counterions are discussed.

Shell formation in short like-charged polyelectrolytes in a harmonic trap.

Inspired by recent experiments and simulations on pattern formation in biomolecules by optical tweezers, a theoretical description based on the reference interaction site model (RISM) is developed to calculate the equilibrium density profiles of small polyelectrolytes in an external potential. The formalism is applied to the specific case of a finite number of Gaussian and rodlike polyelectrolytes trapped in a harmonic potential. The density profiles of the polyelectrolytes are studied over a range of lengths and numbers of polyelectrolytes in the trap, and the strength of the trap potential. For smaller polymers we recover the results for point charges. In the mean field limit the longer polymers, unlike point charges, form a shell at the boundary layer. When the interpolymer correlations are included, the density profiles of the polymers show sharp shells even at weaker trap strengths. The implications of these results are discussed.

Green Function of Correlated Genes and the Mechanical Evolution of Protein

There has been growing evidence that cooperative interactions and configurational rearrangements underpin protein functions. But in spite of vast genetic and structural data, the information-dense, heterogeneous nature of protein has held back the progress in understanding the underlying principles. Here we outline a general theory of protein that quantitatively links sequence, dynamics and function: The protein is a strongly-coupled amino acid network whose interactions and large-scale motions are captured by the mechanical propagator, also known as the Green function. The propagator relates the gene to the connectivity of the amino acid network and the transmission of forces through the protein. How well the force pattern conforms to the collective modes of the functional protein is measured by the fitness. Mutations introduce localized perturbations to the propagator which scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. Epistasis quantifies how much the combined effect of multiple mutations departs from additivity. We find that epistasis is the nonlinearity of the Green function, which corresponds to a sum over multiple scattering paths passing through the localized perturbations. We apply this mechanical framework to the simulations of protein evolution, and observe long-range epistasis which facilitates collective functional modes. Our model lays the foundation for understanding the protein as an evolved state of matter and may be a prototype for other strongly-correlated living systems.