Components Of Pressure Tensor Research Articles

Conditions for the formation of nongyrotropic current sheets in slowly evolving plasmas

This paper addresses the formation of nongyrotropic current sheets resulting from slow external driving. The medium is a collisionless plasma with one spatial dimension and a three-dimensional velocity space. The study is based on particle simulation and an analytical approach. Earlier results that apply to compression of an initial Harris sheet are generalized in several ways. In a first step a general sufficient criterion for the presence of extra ion and electron currents due to nongyrotropic plasma conditions is derived. Then cases with antisymmetric magnetic and electric fields are considered. After establishing consistency of the criterion with the earlier case, the usefulness of this concept is illustrated in detail by two further particle simulations. The results indicate that the formation of nongyrotropic current sheets is a ubiquitous phenomenon for plasmas with antisymmetric fields that have evolved slowly from initial gyrotropic states. A fourth case concerns a plasma with a unidirectional magnetic field. Consistent with the general criterion, the observed final state is fluidlike in that it is approximately gyrotropic. Momentum balance is shown to include a contribution that results from accumulation of an off-diagonal pressure tensor component during the evolution. Heat flux also plays an important role.

Disjoining pressure at the edge of plane-parallel slit between solids interacting by dispersion forces

The approach to calculating the Irving-Kirkwood pressure tensor inside plane-parallel hollow slit between solids interacting by dispersion forces is developed. Disjoining pressure is defined as a normal component of pressure tensor at slit walls and is calculated as explicit function of both the width of slit and distance from its edge. It is revealed that, at preset slit width, disjoining pressure acquires its common value for an infinite slit at triple (by the width) distance from the slit edge.

A theory of a curved vapor-liquid separation boundary: The lattice gas model

Molecular theory of curved vapor-liquid interphase boundaries was considered in terms of the lattice gas model. The theory uses the quasi-thermodynamic concept of curved layers of a separation boundary with a large radius. The transition from a rectangular lattice to such layers is performed by the introduction of a variable number of the nearest neighbors. The problems (1) of the transition from distributed molecular models to layer models reflecting macroscopic symmetry of the interphase boundary and (2) of a minimum linear size of the surface region to which thermodynamic approaches are applicable were considered. Equations for the quasi-equilibrium distribution of molecules at the vapor-liquid boundary in a metastable system were constructed in the quasi-chemical approximation taking into account direct correlations between the nearest interacting molecules. A metastable state is maintained by a pressure jump described by the macro-scopic Laplace equation on a separation surface inside the interphase region. Equations for local mean pressure values and normal and tangential pressure tensor components inside the interphase region were constructed. These equations were used to obtain microscopic difference mechanical equilibrium equations for curved boundaries of spherical and cylindrical drops in the metastable state. The relation between the micro-scopic difference mechanical equilibrium equations and similar differential equations and the macroscopic Laplace equation, which described pressure jump in a metastable system, was considered. Various definitions of surface tension are discussed.

Quasi-2D and prewetting transitions of square-well fluids on a square-well substrate

Molecular simulation methodologies are employed to study the first-order transition of variable square-well (SW) fluids on a wide range of weak attractive surfaces. Surface phase diagram of SW fluids of attractive well diameter λ ff = 1.5, 1.75, 2.0 on a smooth, structureless surface modelled by a SW potential is reported via grand-canonical transition-matrix Monte Carlo (GC-TMMC) and histogram reweighting techniques. Fluids with λ ff = 1.5 and 1.75 show quasi-2D vapour–liquid phase transition; on the other hand, prewetting transition is visible for a SW fluid with larger well-extent λ ff = 2.0. The prewetting line, its length, and closeness to the bulk saturation curve are found to depend strongly on the nature of the fluid–fluid and fluid–wall interaction potentials. Boundary tension of surface coexistence films is calculated by two methods. First, the finite size scaling approach of Binder is used to evaluate the boundary tension via GC-TMMC. Second, the results of the boundary tension are verified by virtue of its relation to the pressure tensor components, which are calculated using a NVT-Monte Carlo approach. The results from the two methods are in good agreement. Boundary tension is found to increase with the increase in the wall–fluid interaction range for the quasi-2D system; conversely, boundary tension for thin–thick film, at prewetting transition, decreases with the increase in the wall–fluid interaction range.

Phase behavior and fluid–solid surface tension of argon in slit pores and carbon nanotubes

Phase behaviors of argon in several types of cylindrical and slit pores are examined by grand-canonical Monte Carlo simulations. Condensation processes in single- and multi-walled carbon nanotubes along with those in hard-wall tubes are compared. Effects of the pore size on pressure–tensor components, the fluid–wall surface tension, and the adsorption are also compared for the different fluid-pore interactions. The chemical potential at which the fluid begins to condense in the single-walled nanotube is greater than that in the multi-walled nanotube by an amount nearly equal to the difference in the potential-well depth of the fluid-pore interaction, and the adsorption isotherms overlap each other almost completely for narrow pores and partially for wider pores. Similar analyses are performed for slit pores of two different hydrocarbon models.

Some remarks on one-dimensional force-free Vlasov–Maxwell equilibria

The conditions for the existence of force-free nonrelativistic translationally invariant one-dimensional (1D) Vlasov–Maxwell (VM) equilibria are investigated using general properties of the 1D VM equilibrium problem. As has been shown before, the 1D VM equilibrium equations are equivalent to the motion of a pseudoparticle in a conservative pseudopotential, with the pseudopotential being proportional to one of the diagonal components of the plasma pressure tensor. The basic equations are here derived in a way different from previous work. Based on this theoretical framework, a necessary condition on the pseudopotential (plasma pressure) to allow for force-free 1D VM equilibria is formulated. It is shown that linear force-free 1D VM solutions, which so far are the only force-free 1D VM solutions known, correspond to the case where the pseudopotential is an attractive central potential. A general class of distribution functions leading to central pseudopotentials is discussed.

Phase equilibria and interfacial tension of fluids confined in narrow pores

Correlation between phase behaviors of a Lennard-Jones fluid in and outside a pore is examined over wide thermodynamic conditions by grand canonical Monte Carlo simulations. A pressure tensor component of the confined fluid, a variable controllable in simulation but usually uncontrollable in experiment, is related with the pressure of a bulk homogeneous system in equilibrium with the confined system. Effects of the pore dimensionality, size, and attractive potential on the correlations between thermodynamic properties of the confined and bulk systems are clarified. A fluid-wall interfacial tension defined as an excess grand potential is evaluated as a function of the pore size. It is found that the tension decreases linearly with the inverse of the pore diameter or width.

Structure and dynamics of a new class of thin current sheets

New results on a steady state Vlasov theory of current sheets, which generalizes the Harris (1962) model by assuming anisotropic and nongyrotropic plasmas and using the invariant of particle motion in regions of strong gradients, are presented with the aim to explain multiprobe observations of thin current sheets in the geomagnetotail and laboratory experiments, including the effects of current sheet embedding and bifurcation. The dynamics of these sheets is explored using a full particle code with more realistic mass ratio and anisotropy parameters than those used in our earlier works. The results relevant to 2001 CLUSTER observations, with the sheet thickness appreciably exceeding the thermal ion gyroradius, include ion distributions and pressure tensor components, which reveal the important role of nongyrotropic effects on the structure of these sheets. Their flapping motions are distinguished by north‐south asymmetry of current profiles, quasi‐rectangular shape of the flapping waves, and their small propagation speed, suggesting an explanation of their propagation toward the flanks of the tail sheet. The main effect of the ion anisotropy on the sheets with thickness less than the thermal ion gyroradius, relevant to 2003 CLUSTER observations and laboratory experiments, is their charging, which may limit their minimum thickness, while their structure can be modified by electron anisotropy. Other distinctive features of these sheets are three‐peaked current density profiles, found both in simulations and in the steady state theory, the north‐south asymmetry of flapping sheets, and the shape of flapping waves, which is drastically different from the case of thicker sheets.

Oscillatory surface tension due to finite-size effects

The simulation results of surface tension at the liquid-vapor interface are presented for fluids interacting with Lennard Jones and square-well potentials. From the simulation of liquids we have reported [M. González-Melchor et al., J. Chem. Phys. 122, 4503 (2005)] that the components of pressure tensor in parallelepiped boxes are not the same when periodic boundary conditions and small transversal areas are used. This fact creates an artificial oscillatory stress anisotropy in the system with even negative values. By doing direct simulations of interfaces we show in this work that surface tension has also an oscillatory decay at small surface areas; this behavior is opposite to the monotonic decay reported previously for the Lennard Jones fluid. It is shown that for small surface areas, the surface tension of the square-well potential artificially takes negative values and even increases with temperature. The calculated surface tension using a direct simulation of interfaces might have two contributions: one from finite-size effects of interfacial areas due to box geometry and another from the interface. Thus, it is difficult to evaluate the true surface tension of an interface when small surface areas are used. Care has to be taken to use the direct simulation method of interfaces to evaluate the predicted surface tension as a function of interfacial area from capillary-wave theory. The oscillations of surface tension decay faster at temperatures close to the critical point. It is also discussed that a surface area does not show any important effect on coexisting densities, making this method reliable to calculate bulk coexisting properties using small systems.

Methodological problems in pressure profile calculations for lipid bilayers

From molecular dynamics simulations of a dipalmitoyl-phosphatidyl-choline (DPPC) lipid bilayer in the liquid crystalline phase, pressure profiles through the bilayer are calculated by different methods. These profiles allow us to address two central and unresolved problems in pressure profile calculations: The first problem is that the pressure profile is not uniquely defined since the expression for the local pressure involves an arbitrary choice of an integration contour. We have investigated two different choices leading to the Irving-Kirkwood (IK) and Harasima (H) expressions for the local pressure tensor. For these choices we find that the pressure profile is almost independent of the contour used, which indicates that the local pressure is well defined for a DPPC bilayer in the liquid crystalline phase. This may not be the case for other systems and we therefore suggest that both the IK and H profiles are calculated in order to test the uniqueness of the profile. The second problem is how to include electrostatic interactions in pressure profile calculations when the simulations are conducted without truncating the electrostatic potential, i.e., using the Ewald summation technique. Based on the H expression for the local pressure, we present a method for calculating the contribution to the lateral components of the local pressure tensor from electrostatic interactions evaluated by the Ewald summation technique. Pressure profiles calculated with an electrostatic potential truncation (cutoff) from simulations conducted with Ewald summation are shown to depend on the cutoff in a subtle manner which is attributed to the existence of long-ranged charge ordering in the system. However, the pressure profiles calculated with relatively long cutoffs are qualitatively similar to the Ewald profile for the DPPC bilayer studied here.

A gyrokinetic electron and fully kinetic ion plasma simulation model

A novel new kinetic simulation model has been developed to investigate dynamics in collisionless plasmas. In this model, the electrons are treated as gyrokinetic (GK) particles and ions are treated as fully kinetic (FK) particles. In the GK-electron and FK-ion (GKe/FKi) plasma simulation model, the rapid electron cyclotron motion is removed, while keeping finite electron Larmor radii, realistic electron-to-ion mass ratio, wave–particle interactions, and off-diagonal components of the electron pressure tensor. The model is particularly suitable for plasma dynamics with wave frequencies lower than the electron gyrofrequency, and for problems in which the wave modes ranging from Alfvén waves to lower-hybrid/whistler waves need to be handled on an equal footing. Using this model, the computation power can be significantly improved over that of the existing full-particle codes. The GKe/FKi model, furthermore, can also handle physics with realistic electron-to-ion mass ratio and dynamic processes on the global Alfvén time/spatial scales. With respect to the hybrid (i.e. FK ion and fluid electron) model, the GKe/FKi model has the advantage that important electron kinetic physics, such as wave–particle resonances and finite electron Larmor radius effects, are included. The simulation model has been successfully benchmarked for linear waves in uniform plasmas against analytic dispersion relation.

Structural changes and viscoplastic behavior of a generic embedded-atom model metal in steady shear flow.

We study equilibrium and nonequilibrium properties of a simple "generic embedded-atom model" (GEAM) for metals. The model allows to derive simple analytical expressions for several zero-temperature constitutive properties--in overall agreement with real metals. The model metal is then subjected to shear deformation and strong flow via nonequilibrium molecular dynamics simulation in order to discuss the origins of some qualitative properties observed using more specific embedded-atom potentials. The "common neighbor analysis," based on planar graphs is used to obtain information about the transient structures accompanying viscoplastic behavior on an atomic level. In particular, pressure tensor components and plastic yield are investigated and correlated with underlying structural changes. A simple analytical expression for the isotropic pressure at finite temperatures is proposed. A nonequilibrium phase diagram is obtained by semianalytic calculation.

Surface tension of a square well fluid

We performed Monte Carlo simulations in the canonical ensemble on the liquid–vapor interface of a square well fluid with interaction range of λ=1.5σ. The system contains a liquid slab surrounded by vapor. The surface tension is calculated during simulations by using an original procedure that allows the calculation of the pressure tensor components. The surface tension decreases monotonically with temperature. Coexisting densities and pressure along the liquid–vapor coexistence line have also been obtained and good agreement is found with results calculated from bulk simulations.

Collisionless reconnection supported by nongyrotropic pressure effects in hybrid and particle simulations

This paper presents the detailed comparative analysis of full particle and hybrid simulations of collisionless magnetic reconnection. The comprehensive hybrid simulation code employed in this study incorporates essential electron kinetics in terms of the evolution of the full electron pressure tensor in addition to the full ion kinetics and electron bulk flow inertia effects. As was demonstrated in our previous publications, the electron nongyrotropic pressure effects play the dominant role in supporting the reconnection electric field in the immediate vicinity of the neutral X point. The simulation parameters are chosen to match those of the Geospace Environmental Modeling (GEM) “Reconnection Challenge.” It is that these comprehensive hybrid simulations perfectly reproduce the results of full particle simulations in many details. Specifically, the time evolutions of the reconnected magnetic flux and the reconnection electric field, as well as spatial distributions of current density and magnetic field at all stages of the reconnection process, are found to be nearly identical for both simulations. Comparisons of variations of characteristic quantities along the x and z axes centered around the dominating X points also revealed a remarkable agreement. Noticeable differences are found only in electron temperature profiles, i.e., in the diagonal electron pressure tensor components. The deviation in the electron heating pattern in hybrid simulations from that observed in particle simulations, however, does not affect parameters essential for the reconnection process. In particular, the profiles of the off‐diagonal components of the electron pressure tensor are found to be very similar for both runs and appear unaffected by heat flux effects. Both simulations also demonstrate that the Ey component of the electric field is nearly constant inside the diffusion region where ions are nonmagnetized. We demonstrate that the simple analytical estimate for the reconnection electric field as a convection electric field at the edge of the diffusion region very well reproduces the reconnection electric field observed in the simulations.

The diffusion region in collisionless magnetic reconnection

The structure of the dissipation region in collisionless magnetic reconnection is investigated by means of kinetic particle-in-cell simulations and analytical theory. Analyses of simulations of reconnecting current sheets without guide magnetic field, which keep all parameters fixed with the exception of the electron mass, exhibit very similar large scale evolutions and time scales. A detailed comparison of two runs with different electron masses reveals very similar large scale parameters, such as ion flow velocities and magnetic field structures. The electron-scale phenomena in the reconnection region proper, however, appear to be quite different. The scale lengths of these processes are best organized by the trapping length of bouncing electrons in a field reversal region. The dissipation is explained by the electric field generated by nongyrotropic electron pressure tensor effects. In the reconnection region, the relevant electron pressure tensor components exhibit gradients which are independent of the electron mass. The similarities of the gradients as well as the behavior of the electron flow velocity can be derived from the electron trapping scale and the electron mass independence of the reconnection electric field. A further model which includes a significant guide magnetic field exhibits almost identical behavior. The explanation of this result lies in a Hall-type electric field which locally eliminates the magnetizing effect on the electrons of the guide magnetic field. The resulting electron dynamics is nearly identical to the one found in the model without guide magnetic field. This result strongly supports the hypothesis that the local physics in the dissipation region adjusts itself to the demands of the large-scale evolution. A further verification of this notion is provided by Hall-magnetohydrodynamic simulations which employ simple resistive dissipation models in otherwise similar large-scale models. These results also pertain to the inclusion of local reconnection physics in larger scale simulation models.

Density functional theory for spherical drops

We have applied Sullivan's model to study the interfacial properties of a spherical drop embedded in a one-component fluid using the mean field approximation. We have examined the effect of drop size (expressed by the equimolar radius Re) on the density profile, pressure tensor, pressure difference Delta p across the interface, surface tension, Tolman's length, and density, pressure, and normal component of pressure tensor at the centre of the drop. Delta p is found to be an ill defined quantity in the sense that it can be defined in various ways, whose results coincide for large drops. The surface tension is a non-monotonic function of Re; it increases slowly from its flat interface value as Re decreases until a maximum value is attained, then it decreases rapidly. For small supersaturations, the drops are under tension and compression while at large supersaturations they are only under tension. The results of this theory are compared qualitatively with previous molecular dynamics simulations and theoretical calculations for drops.

Comment on ‘‘Molecular dynamics of the surface tension of a drop’’ [J. Chem. Phys. 9 6, 565 (1992)

The procedure of the estimation of Tolman’s length δ for small droplets based only on the normal component of the local pressure tensor is proposed. The treatment of the computer simulation data with the help of this method results in the definite positive value of δ for the Lennard-Jones droplets.

Supersonic lattice gases: Restoration of Galilean invariance by nonlinear resonance effects

Energy-conserving multispeed lattice simulations of fluid behavior are shown to be capable of maintaining approximate macroscopic Galilean invariance at supersonic flow velocities. This result is due to a resonance between macroscopic flow velocity and discrete microscopic velocities, a nonlinear effect which was suppressed in previous power series expansion derivations of the macroscopic flow equations. The effect is demonstrated by numerical evaluation of the pressure tensor components for two-dimensional lattices with 25, 57, and 129 bits per lattice site and by simulations of a two-dimensional free shear decay flow up to Mach 1.9.

Self‐consistent plasma pressure tensors from the Tsyganenko Magnetic Field Models

The high‐pressure plasma trapped in the geomagnetic tail strongly shapes the surrounding magnetic field. Force balance in a steady state requires a balance between the divergence of the total particle momentum stress tensor and the divergence of the magnetic field momentum stress tensor which is the Lorentz force. This condition of stress balance allows strict constraints on the total particle pressure tensor to be derived from the magnetic field structure. The Kp parameterized empirical magnetospheric fields of the Tsyganenko model with local two‐dimensional approximations are used to derive the pressure tensor components implied for the central plasma sheet. The small anisotropies p⊥/p∥ in the gyrotropic pressure tensor required for equilibrium in the central plasma sheet are calculated from the nonpotential values found for the j × B force. The required ratio p⊥/p∥ approaches the marginal stability criterion for the magnetohydrodynamic (MHD) mirror mode and the Alfvén ion cyclotron mode, and we suggest that these instabilities may play a role in establishing the equilibrium structure of the geomagnetic tail. From this perspective the equilibrium and stability constraints for the maximum allowed anisotropy suggest that the Tsyganenko magnetotail field parameterization needs to be modified to support self‐consistent gyrotropic equilibria.

Numerical simulation of satellite-ring interactions: Resonances and satellite-ring torques

A kinetic equation is solved numerically for a flattened planetary ring which is perturbed gravitationally by a near-by satellite. The Krook kinetic equation for planetary rings is solved in two spatial dimensions, and in time, with interparticle collisions, with satellite forcing, and without self-gravity. The phase-space fluid numerical method is extended to two-dimensional systems by ignoring negligible high-order velocity moments. In simulations of satellite induced wakes, we verify the role of local shear reversal and angular momentum flux decrease as described by Borderies et al. (1983, Icarus 55, 124–132). A new result is that the amplitude of wakes is limited by purely kinematic effects, even in the absence of collisions. The results of a simulation of an inner Lindblad resonance location, as the distribution approaches steady-state, are presented in detail. The surface mass density, pressure tensor components, and mean velocities are calculated during the simulation. The detailed mechanisms of local torque balance and forced eccentricity “damping” at resonance are illuminated. The angular momentum flux perturbations play the critical role of balancing the satellite torque on the surface mass density asymmetry. The eccentricities very close to resonance are limited by collisional phase mixing. The resulting near steady-state distribution feels a net radially integrated torque of essentially zero from the satellite to the first-order accuracy intended by the method, though there are large negative and positive torques on the regions just inside of and outside of the exact resonance. We show that previous analytic calculations of the torque (which show nonzero net torque) were the result of small second-order deviations of the surface density on top of the large first-order structure which our simulations reproduce, while there are second-order effects not accounted for in those calculations. Our simulations do not show an increase of the velocity dispersion in the resonance zone, so that energy conservation considerations do not require a net torque due to conservation of the Jacobi constant of the planar restricted three-body problem. We therefore argue that simulations or calculations which include all second-order effects are needed in order to determine the torques which will actually occur in rings.