FENE stands for the finitely extensible nonlinear elastic model of a long-chained polymer. It simplifies the chain of monomers by connecting a sequence of beads with nonlinear springs. The spring force law is governed by inverse Langevin function or approximated by the Warner's relationship
Where and H is the spring constant.
Total stretching force on ith bead can be written as: .
Statics is the branch of mechanics that is concerned with the analysis of loads acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives:
Kinematics is a subfield of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively.
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in both science and engineering. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy.
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
The NTRUEncrypt public key cryptosystem, also known as the NTRU encryption algorithm, is a lattice-based alternative to RSA and ECC and is based on the shortest vector problem in a lattice.
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables.
In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.
An ideal chain is the simplest model to describe polymers, such as nucleic acids and proteins. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers. Although it is simple, its generality gives insight about the physics of polymers.
The worm-like chain (WLC) model in polymer physics is used to describe the behavior of polymers that are semi-flexible: fairly stiff with successive segments pointing in roughly the same direction, and with persistence length within a few orders of magnitude of the polymer length. The WLC model is the continuous version of the Kratky–Porod model.
In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling.
A shear band is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials. As an example, a soil specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized.
rubber elasticity refers to the remarkable property of crosslinked rubber: it can be stretched by up to a factor of 10 from its original length and, when released, returns very nearly to its original length. This can be repeated many times with no apparent degradation to the rubber. Rubber is a member of a larger class of materials call elastomers and it is difficult to overestimate their economic and technological importance. Elastomers have played a key role in the development of new technologies in the 20th century and make a substantial contribution to the global economy. Water-tight seals, latex gloves and tires are examples of elastomer-enabled materials that are now crucial to modern society. Rubber elasticity is a complex process. It is fundamentally different from the restoring force in a spring, which relies on the mechanical stiffness of the material. In rubber the elastic force is due to the thermal interactions of the molecules within the material. There are several molecular mechanisms that work together to produce the elastic force. Understanding the physics of elasticity requires a knowledge of advanced mathematics, chemistry and statistical physics, particularly the concept of entropy. Entropy may be thought of as a measure of the thermal energy that is stored in a molecule. Common rubbers, such as polybutadiene and polyisoprene, are produced by a process called polymerization. Very long molecules (polymers) are built up sequentially by adding short molecular backbone units through chemical reactions. The backbone unit for rubber is a short chain composed of four carbon atoms. The chemical bonds that connect the carbon atoms to one another do not form a straight line and usually four sequential carbon atoms do not even lie in a plane. The result is that a rubber polymer follows a random, zigzag path in three dimensions, intermingling with many other rubber molecules. The polymer chain may be regarded as a series of local kinks, each composed of a few backbone units. To produce an elastomer from the long rubber molecules, a molecular network must be created. This is accomplished by the addition of a few percent of a cross linking molecule such as sulfur. When heated, the crosslinking molecule causes a reaction that chemically joins (bonds) two of the rubber molecules together at some point. Because the rubber molecules are so long, each one participates in many crosslinks with many other rubber molecules. The polymer segments between two crosslinks are called network chains. The crosslinks transform all of the initially separate polymers into a single enormous molecule. A rubber band is a single molecule as is a latex glove! When a rubber band is stretched, some of the network chains are forced to become straight. Individual 4-carbon backbone units are forced into rotational conformations that have longer end-to-end distances and this restricts the number of conformations (states) that are thermally accessible. Fewer available conformational states means that a stretched backbone unit can store less thermal energy. Hence its entropy must decrease. It is this decrease in entropy that gives rise to the elastic force in the network chains.
The Gaussian network model (GNM) is a representation of a biological macromolecule as an elastic mass-and-spring network to study, understand, and characterize the mechanical aspects of its long-time large-scale dynamics. The model has a wide range of applications from small proteins such as enzymes composed of a single domain, to large macromolecular assemblies such as a ribosome or a viral capsid. Protein domain dynamics plays key roles in a multitude of molecular recognition and cell signalling processes. Protein domains, connected by intrinsically disordered flexible linker domains, induce long-range allostery via protein domain dynamics. The resultant dynamic modes cannot be generally predicted from static structures of either the entire protein or individual domains.
FENE-P is a continuous model of polymer. The name FENE stands for finitely extensible nonlinear elastic while P stands for the closure proposed by Peterlin. It takes the dumbbell version of the FENE model and assumed the Peterline statistical closure for the restoring force.
Reptation is the thermal motion of very long linear, entangled macromolecules in polymer melts or concentrated polymer solutions. Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another. Pierre-Gilles de Gennes introduced the concept of reptation into polymer physics in 1971 to explain the dependence of the mobility of a macromolecule on its length. Reptation is used as a mechanism to explain viscous flow in an amorphous polymer. Sir Sam Edwards and Masao Doi later refined reptation theory. Similar phenomena also occur in proteins.
In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. Its roots lie in the classical Karhunen (1946)–Loève spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem. SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition.
In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered the de facto standard in the theory of nonlinear state estimation, navigation systems and GPS.
The Rouse model is frequently used in polymer physics.
The Frenkel–Kontorova model, also known as the FK model, is a fundamental model of low-dimensional nonlinear physics.
