Tonks Gas Research Articles

Kinetic temperature and pressure of an active Tonks gas.

Using computer simulation and analytical theory, we study an active analog of the well-known Tonks gas, where active Brownian particles are confined to a periodic one-dimensional (1D) channel. By introducing the notion of a kinetic temperature, we derive an accurate analytical expression for the pressure and clarify the paradoxical behavior where active Brownian particles confined to 1D exhibit anomalous clustering but no motility-induced phase transition. More generally, this work provides a deeper understanding of pressure in active systems as we uncover a unique link between the kinetic temperature and swim pressure valid for active Brownian particles in higher dimensions.

Dynamical many-body delocalization transition of a Tonks gas in a quasi-periodic driving potential

The quantum kicked rotor is well-known for displaying dynamical (Anderson) localization. It has recently been shown that a periodically kicked Tonks gas will always localize and converge to a finite energy steady-state. This steady-state has been described as being effectively thermal with an effective temperature that depends on the parameters of the kick. Here we study a generalization to a quasi-periodic driving with three frequencies which, without interactions, has a metal-insulator Anderson transition. We show that a quasi-periodically kicked Tonks gas goes through a dynamical many-body delocalization transition when the kick strength is increased. The localized phase is still described by a low effective temperature, while the delocalized phase corresponds to an infinite-temperature phase, with the temperature increasing linearly in time. At the critical point, the momentum distribution of the Tonks gas displays different scaling at small and large momenta (contrary to the non-interacting case), signaling a breakdown of the one-parameter scaling theory of localization.

Anomalous phase behavior of quasi-one-dimensional attractive hard rods.

We study a two-state model of attractive hard rods using the transfer matrix method, where the centers of the particles are confined to a straight line, but the orientations of the rods can be parallel or perpendicular to the confining line. The rods are modeled as hard rectangles with length L and width D and decorated with attractive sites at both ends of the rectangles. We find that the particles align parallel to the line and form long chains at low densities, while they turn out of the line and form a Tonks gas at high densities. With increasing the stickiness between the rods, the structural change between parallel and perpendicular states becomes stronger and the pressure vs density curve becomes almost a horizontal line at the transition pressure. We show that such a behavior is reminiscent of the first-order phase transition. This manifests in the validity of the lever rule of the phase transitions for very sticky cases.

Noncooperative thermodynamics and kinetic models of ligand binding to polymers: Connecting McGhee-von Hippel model with the Tonks gas model.

Ligand binding to polymers modifies the physical and chemical properties of the polymers, leading to physical, chemical, and biological implications. McGhee and von Hippel obtained the equilibrium coverage as a function of the ligand affinity, through the computation of the possible binding sites for the ligand. Here, we complete this theory deriving the kinetic model for the ligand-binding dynamics and the associated equilibrium chemical potential, which turns out to be of the Tonks gas model type. At low coverage, the Tonks chemical potential becomes the Fermi chemical potential and even the ideal gas chemical potential. We also discuss kinetic models associated with these chemical potentials. These results clarify the kinetic models of ligand binding, their relations with the chemical potentials, and their range of validity. Our results highlight the inaccuracy of ideal and simplified kinetic approaches for medium and high coverages.

Zeros of partition functions in the NPT ensemble.

Lee-Yang and Fisher zeros are crucial for the study of phase transitions in the grand canonical and the canonical ensembles, respectively. However, these powerful methods do not cover the isothermal-isobaric ensemble (NPT ensemble), which reflects the conditions of many experiments. In this work we present a theory of the phase transitions in terms of the zeros of the NPT-ensemble partition functions in the complex plane. The proposed theory provides an approach to calculate all the partition function zeros in the NPT ensemble, which form certain curves in the thermodynamic limit. To verify the theory we consider Tonks gas and van der Waals fluid in the NPT ensemble. In the case of Tonks gas, similarly to the Lee-Yang circle theorem, we obtain an exact equation for the zero limit curve. We also derive an approximated limit curve equation for van der Waals fluid in terms of the Szegö curve. This curve fits numerically calculated zeros and correctly describes how the phenomenon of phase transition depends on the temperature.

The properties of Tonks–Girardeau gas at finite temperature and comparison with spin-polarized fermions

In the present paper, we investigate the Tonks–Girardeau (TG) gas confined in a harmonic trap at finite temperature with thermal Bose–Fermi mapping method. The pair distribution, density distribution, reduced one-body density matrix (ROBDM), the first-order correlation function, the occupations number of natural orbitals and momentum distribution are evaluated. In the whole temperature regime, the pair distribution and density distribution exhibit the same properties as those of spin-polarized fermions because both of them depend on the modulus of wavefunction rather than wavefunction. While the ROBDM, the first-order correlation function, the natural orbital occupation, momentum distribution, which depend on wavefunction, of Tonks gas displays Bose properties different from spin-polarized fermions at low temperature. At high temperature we cannot distinguish Tonks gas from the spin-polarized Fermi gas qualitatively by all properties.

A one-dimensional statistical mechanics model for nucleosome positioning on genomic DNA

The first level of folding of DNA in eukaryotes is provided by the so-called ‘10 nm chromatin fibre’, where DNA wraps around histone proteins (∼10 nm in size) to form nucleosomes, which go on to create a zig-zagging bead-on-a-string structure. In this work we present a one-dimensional statistical mechanics model to study nucleosome positioning within one such 10 nm fibre. We focus on the case of genomic sheep DNA, and we start from effective potentials valid at infinite dilution and determined from high-resolution in vitro salt dialysis experiments. We study positioning within a polynucleosome chain, and compare the results for genomic DNA to that obtained in the simplest case of homogeneous DNA, where the problem can be mapped to a Tonks gas []. First, we consider the simple, analytically solvable, case where nucleosomes are assumed to be point-like. Then, we perform numerical simulations to gauge the effect of their finite size on the nucleosomal distribution probabilities. Finally we compare nucleosome distributions and simulated nuclease digestion patterns for the two cases (homogeneous and sheep DNA), thereby providing testable predictions of the effect of sequence on experimentally observable quantities in experiments on polynucleosome chromatin fibres reconstituted in vitro.

Reduced one-body density matrix of Tonks–Girardeau gas at finite temperature**Project supported by the National Natural Science Foundation of China (Grant No. 11004007) and the Fundamental Research Funds for the Central Universities of China.

With thermal Bose–Fermi mapping method, we investigate the Tonks–Girardeau gas at finite temperature. It is shown that at low temperature, the Tonks gas displays the Fermi-like density profiles, and with the increase in temperature, the Tonks gas distributes in wider region. The reduced one-body density matrix is diagonal dominant in the whole temperature region, and the off-diagonal elements shall vanish rapidly with the deviation from the diagonal part at high temperature.

Hard rods in a cylindrical pore: the nematic-to-smectic phase transition

The effect of cylindrical confinement on the phase behaviour of a system of parallel hard rods is studied using Onsager’s second-virial theory. The hard rods are represented as hard cylinders of diameter D and length L, while the cylindrical pore is infinite with diameter W. The interaction between the wall and the rods is hard repulsive, and it is assumed that molecules are parallel to the surface of the pore (planar anchoring). In very narrow pores (D < W < 2D), the structure is homogeneous and the system behaves as a one-dimensional Tonks gas. For wider pores, inhomogeneous fluid structures emerge because of the lowering of the average excluded volume due to the wall–particle interaction. The bulk nematic–smectic A phase transition is replaced by a transition between inhomogeneous nematic and smectic A phases. The smectic is destabilized with respect to the nematic for decreasing pore width; this effect becomes substantial for W < 10D. For W > 100D, results for bulk and confined fluids agree well due to the short range effect of the wall (∼3–4D).

Entropic enhancement of spatial correlations in a laser-driven Rydberg gas

In a laser-driven Rydberg gas the strong interaction between atoms excited to Rydberg states results in the formation of collective excitations. Atoms within a so-called blockade volume share a single Rydberg excitation, which is dynamically created and annihilated. For sufficiently long times this driven system approaches a steady state, which lends its properties from a maximum entropy state of a Tonks gas. Using this connection we show that spatial correlations between Rydberg atoms are controlled by the number of atoms contained within a blockade volume. For a small number the system favors a disordered arrangement of Rydberg atoms, whereas in the opposite limit Rydberg atoms tend to arrange in an increasingly ordered configuration. We argue that this is an entropic effect which is observable in current experiments.

Drivenk-mers: Correlations in space and time

Steady-state properties of hard objects with exclusion interaction and a driven motion along a one-dimensional periodic lattice are investigated. The process is a generalization of the asymmetric simple exclusion process (ASEP) to particles of length k, and is called the k-ASEP. Here, we analyze both static and dynamic properties of the k-ASEP. Density correlations are found to display interesting features, such as pronounced oscillations in both space and time, as a consequence of the extended length of the particles. At long times, the density autocorrelation decays exponentially in time, except at a special k-dependent density when it decays as a power law. In the limit of large k at a finite density of occupied sites, the appropriately scaled system reduces to a nonequilibrium generalization of the Tonks gas describing the motion of hard rods along a continuous line. This allows us to obtain in a simple way the known two-particle distribution for the Tonks gas. For large but finite k, we also obtain the leading-order correction to the Tonks result.

Glassy dynamics and hysteresis in a linear system of orientable hard rods

We study the dynamics of a one-dimensional fluid of orientable hard rectangles with a non-coarse-grained microscopic mechanism of facilitation. The length occupied by a rectangle depends on its orientation, which is a discrete variable coupled to an external field. The equilibrium properties of our model are essentially those of the Tonks gas, but at high densities the orientational degrees of freedom become effectively frozen due to jamming. This is a simple analytically tractable model of the glassy phase. Under a cyclic variation of the pressure, hysteresis is observed. Following a pressure quench, the orientational persistence exhibits a two-stage decay characteristic of glassy systems.

Composite-fermionization of the mixture composed of Tonks gas and Fermi gas

This paper investigates the ground-state properties of the mixture composed of the strongly interacting Tonks—Girardeau gas and spin polarized Fermi gas confined in one-dimensional harmonic traps, where the interaction between the Bose atoms and Fermi atoms is tunable. With a generalized Bose—Fermi transformation the mixture is mapped into a two-component Fermi gas. The homogeneous Fermi gas is exactly solvable by the Bethe-ansatz method and the ground state energy density can be obtained. Combining the ground-state energy function of the homogeneous system with local density approximation it obtains the ground-state density distributions of inhomogeneous mixture. It is shown that with the increase in boson—fermion interaction, the system exhibits composite-fermionization crossover.

Ion-induced density bubble in a strongly correlated one-dimensional gas

We consider a harmonically trapped Tonks-Girardeau gas of impenetrable bosons in the presence of a single embedded ion, which is assumed to be tightly confined in an RF trap. In an ultracold ion-atom collision the ion's charge induces an electric dipole moment in the atoms which leads to an attractive ${r}^{\ensuremath{-}4}$ potential asymptotically. We treat the ion as a static deformation of the harmonic trap potential and model its short range interaction with the gas in the framework of quantum defect theory. The molecular bound states of the ionic potential are not populated due to the lack of any possible relaxation process in the Tonks-Girardeau regime. Armed with this knowledge we calculate the density profile of the gas in the presence of a central ionic impurity and show that a density bubble of the order of $1\ensuremath{\mu}$m occurs around the ion for typical experimental parameters. From these exact results we show that an ionic impurity in a Tonks gas can be described using a pseudopotential approximation, allowing for significantly easier treatment.

Nucleosome Organization is Quantitatively Described by Statistical Positioning Up- and Downstream of Transcription Start Sites

The positions of nucleosomes in eukaryotic genomes determine which parts of the DNA sequence are readily accessible for regulatory proteins and which parts are not. A salient feature in recent genome-wide nucleosome maps is that nucleosomes appear well-positioned around a nucleosome free region (NFR) just upstream from the transcription start site (TSS). What determines this nucleosome organization is not known. One scenario is that the majority of nucleosome positions near the TSS are directly encoded in the DNA sequence. The alternative “statistical positioning” scenario, is that a few local barriers on the genome strongly constrain the positions of closeby nucleosomes, purely on statistical grounds. We use a physical model for the latter scenario, based on the Tonks gas of statistical physics, to quantitatively analyze recent data for yeast. We find that although the typical patterns on the two sides of the TSS are different, they are both quantitatively described by the same physical model, with the same parameters, but different boundary conditions. The inferred boundary conditions suggest that the first nucleosome downstream from the NFR is typically directly positioned while the first nucleosome upstream is statistically positioned via a nucleosome-repelling DNA region.

Adhesion of surfaces via particle adsorption: exact results for a lattice of fluidcolumns

We present here exact results for a one-dimensional gas, or fluid, of hard-sphere particleswith attractive boundaries. The particles, which can exchange with a bulk reservoir,mediate an interaction between the boundaries. A two-dimensional lattice of suchone-dimensional gas columns represents a discrete approximation of a three-dimensionalgas of particles between two surfaces. The effective particle-mediated interaction potentialof the boundaries, or surfaces, is calculated from the grand-canonical partition function ofthe one-dimensional gas of particles, which is an extension of the well-studied Tonks gas(Tonks (1936 Phys. Rev. 50 955)). The effective interaction potential exhibits two minima.The first minimum at boundary contact reflects depletion interactions, while the secondminimum at separations close to the particle diameter results from a single adsorbedparticle that crosslinks the two boundaries. The second minimum is the global minimumfor sufficiently large binding energies of the particles. Interestingly, the effective adhesionenergy corresponding to this minimum is maximal at intermediate concentrations of theparticles.

Tunneling Exponents Sensitive to Impurity Scattering in Quantum Wires

We show that the scaling exponent for tunneling into a quantum wire in the "Coulomb Tonks gas" regime of impenetrable, but otherwise free, electrons is affected by impurity scattering in the wire. The exponent for tunneling into such a wire thus depends on the conductance through the wire. This striking effect originates from a many-body scattering resonance reminiscent of the Kondo effect. The predicted anomalous scaling is stable against weak perturbations of the ideal Tonks gas limit at sufficiently high energies, similar to the phenomenology of a quantum critical point.

Quantum Many-Body Culling: Production of a Definite Number of Ground-State Atoms in a Bose-Einstein Condensate

We propose a method to produce a definite number of ground-state atoms by adiabatic reduction of the depth of a potential well that confines a degenerate Bose gas with repulsive interactions. Using a variety of methods, we map out the maximum number of particles that can be supported by the well as a function of the well depth and interaction strength, covering the limiting case of a Tonks gas as well as the mean-field regime. We also estimate the time scales for adiabaticity and discuss the recent observation of atomic number squeezing [Chuu, Phys. Rev. Lett. 95, 260403 (2005)10.1103/PhysRevLett.95.260403].

Shock waves in a one-dimensional Bose gas: From a Bose-Einstein condensate to a Tonks gas

We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in shock wave problems. We show that quantum corrections to dynamics of shocks (dark-shock-originated solitons) in a Bose-Einstein condensate are negligible (important) for a realistic set of system parameters. We point out possible signatures of a Bose-Einstein condensate -- Tonks crossover in shock dynamics. Our findings can be directly verified in different experimental setups.

Specific heat of bosons in a lattice

We present a theoretical study of specific heat of bosons ( C v ) in a simple cubic lattice. We have studied the non-interacting bosons and the Tonks gas. For both cases, the C v above the Bose condensation temperature shows considerable temperature dependence compared to that of free bosons. For Tonks gas, we find that the low-temperature specific heat increases as the system gets closer to the Mott transition.