Directions In Configuration Space Research Articles

A new view on vacuum stability in the MSSM

A consistent theoretical description of physics at high energies requires an assessment of vacuum stability in either the Standard Model or any extension of it. Especially supersymmetric extensions allow for several vacua and the choice of the desired electroweak one gives strong constraints on the parameter space. As the general parameter space in the Minimal Supersymmetric Standard Model is huge, any severe constraint on it unrelated to direct phenomenological observations enhances the predictability of the model. We perform an updated analysis of possible charge and color breaking minima without relying on fixed directions in field space that minimize certain terms in the potential (known as "D-flat" directions). Concerning the cosmological stability of false vacua, we argue that there are always directions in configuration space which lead to very short-lived vacua and therefore such exclusions are strict. In addition to existing strong constraints on the parameter space, we find even stronger constraints extending the field space compared to previous analyses and combine those constraints with predictions for the light CP-even Higgs mass in the Minimal Supersymmetric Standard Model. Low masses for supersymmetric partners are excluded from vacuum stability in combination with the 125 GeV Higgs and the allowed parameter space opens at a few TeV.

What Can Systems Theory of Networks Offer to Biology?

What Can Systems Theory of Networks Offer to Biology?

Dynamics of supercooled water in configuration space.

We study the potential energy surface (PES) sampled by a liquid modeled via the widely studied extended simple point charge (SPC/E) model for water. We characterize the curvature of the PES by calculating the instantaneous normal mode (INM) spectrum for a wide range of densities and temperatures. We discuss the information contained in the INM density of states, which requires additional processing to be unambiguously associated with the long-time dynamics. For the SPC/E model, we find that the slowing down of the dynamics in the supercooled region-where the ideal mode coupling theory has been used to describe the dynamics-is controlled by the reduction in the number of directions in configuration space that allow a structural change. We find that the fraction f(dw) of the double-well directions in configuration space determines the value of the diffusion constant D, thereby relating a property of the PES to a macroscopic dynamic quantity; specifically, it appears that square root D is approximately linear in f(dw). Our findings are consistent with the hypothesis that, at the mode coupling crossover temperature, dynamical processes based on the free exploration of configuration space vanish, and processes requiring activation dominate. Hence, the reduction of the number of directions allowing free exploration of configuration space is the mechanism of diffusion implicitly implemented in the ideal mode coupling theory. Additionally, we find a direct relationship between the number of basins sampled by the system and the number of free directions. In this picture, diffusion appears to be related to geometrical properties of the PES, and to be entropic in origin.

Role of unstable directions in the equilibrium and aging dynamics of supercooled liquids

The connectivity of the potential energy landscape in supercooled atomic liquids is investigated through a calculation of the instantaneous normal modes spectrum and a detailed analysis of the unstable directions in configuration space. We confirm the hypothesis that the mode-coupling critical temperature is the T at which the dynamics crosses over from free to activated exploration of configuration space. We also observe changes in the local connectivity of configuration space sampled during aging, following a temperature jump from a liquid to a glassy state.

Instantaneous normal mode analysis of supercooled water

We use the instantaneous normal mode approach to provide a description of the local curvature of the potential energy surface of a model for water. We focus on the region of the phase diagram in which the dynamics may be described by mode-coupling theory. We find that the diffusion constant depends on the fraction of directions in configuration space connecting different local minima, supporting the hypothesis that the dynamics are controlled by the geometric properties of configuration space. Furthermore, we find a relation between the number of basins accessed in equilibrium and the connectivity between them.

Phase mixing in time-independent Hamiltonian systems

Everything you ever wanted to know about what has come to be known as ``chaotic mixing:'' This paper describes the evolution of localised ensembles of initial conditions in 2- and 3-D time-independent potentials which admit both regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) phase space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t) for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined'' chaotic orbits in 2-D systems not stuck near islands by cantori, the moments evolve exponentially: Quantities like the dispersion in p_x, which start small and eventually asymptote towards a larger value, initially grow exponentially in time at a rate comparable to the largest short time Lyapunov exponent. Quantities like | |, that can start large but eventually asymptote towards zero, decrease exponentially. With respect to a discrete L^p norm, reduced distributions f(t) generated from successive decay exponentially towards a near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit considerable structure. Regular ensembles behave very differently, both moments and Df evolving in a fashion better represented by a power law time dependence. ``Confined'' chaotic orbits, initially stuck near regular islands because of cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved in 3-D potentials is qualitatively similar, except that, in this case, it is relatively likely to find one direction in configuration space which is ``less chaotic'' than the other two, so that quantities like L_{ab} depend more sensitively on which phase space variables one tracks.

Classical aspects of quantum spin chains (abstract)

A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.

Some Experimental Results Using Heuristics for Solving the Find-Path Problem in C-Space

The problem of moving a rigid planar body in a 2D environment with fixed obstacles is discussed. In spite of being a simple instance of the general find-path problem, the computational complexity of the existing algorithms to solve it are still too high for practical purposes. A new approach improving the efficiency of the solution to this problem, when viewed as a search problem in configuration space, is introduced. It is based on the use of a reduced model of the connectivity of free-space to globally guide the search process towards the more promising directions in configuration space. The search time savings obtained when using this heuristic are shown to be important and justify its interest for real time applications. Some examples comparing the efficiency of classical search methods with the efficiency of the proposed search procedure are included.

Dynamical Theory of Diffusion. II. Comparison with Rate Theory and the Impurity Isotope Effect

The formalism of the dynamical theory of diffusion is shown to be strictly analogous to that of the reaction-rate theory in the sense that expressions for the jump rate and related quantities can be written in the same form as the corresponding expressions of rate theory. However, the physical meaning of corresponding quantities is different. For example, the role of the saddle-point configuration in rate theory is taken by a dynamical state in the dynamical theory. The reaction coordinate is shown to measure the extent to which lattice reaction forces barring the jump have been overcome by thermal fluctuations. Geometric considerations indicate that for vacancy diffusion, the factor $\ensuremath{\Delta}K$ should have a lower limit on the order of 0.5 in the fcc lattice and an upper limit of the same order for the bcc lattice. In general, $\ensuremath{\Delta}K$ measures the mass dependence of the rms velocity of the system along the jump direction in configuration space. The mass dependence of the isotope effect for impurity diffusion is developed, and a double-isotope-effect experiment involving the simultaneous diffusion of three solute isotopes is proposed to determine impurity correlation factors.