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Linear Response
A linear response function describes the input-output relationship of a signal transducer, such as a radio turning electromagnetic waves into music or a neuron turning Synapse, synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as Magnetic susceptibility, susceptibility, impulse response or Electrical impedance, impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related. Mathematical definition Denote the input of a system by h(t) (e.g. a force), and the response of the system by x(t) (e.g. a position). Generally, the value of x(t) will depend not only on the present value of h(t), but also on past values. Approximately x(t) is a weighted sum of the previous values of h(t'), with the weights given by the linear response function \chi(t-t'): x(t) = \int_^t d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Signal Transducer
Signal transduction is the process by which a chemical or physical signal is transmitted through a cell as a biochemical cascade, series of molecular events. Proteins responsible for detecting stimuli are generally termed receptor (biology), receptors, although in some cases the term sensor is used. The changes elicited by ligand (biochemistry), ligand binding (or signal sensing) in a receptor give rise to a biochemical cascade, which is a chain of biochemical events known as a Cell signaling#Signaling pathways, signaling pathway. When signaling pathways interact with one another they form networks, which allow cellular responses to be coordinated, often by combinatorial signaling events. At the molecular level, such responses include changes in the transcription (biology), transcription or translation (biology), translation of genes, and post-translational modification, post-translational and conformational changes in proteins, as well as changes in their location. These molecula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (that is, if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Susceptibility (other)
Susceptibility may refer to: Physics and engineering In physics the susceptibility is a quantification for the change of an extensive property under variation of an intensive property. The word may refer to: * In physics, the susceptibility of a material or substance describes its response to an applied field. For example: ** Magnetic susceptibility ** Electric susceptibility * The two types of susceptibility above are examples of a linear response function; sometimes the terms ''susceptibility'' and ''linear response function'' are used interchangeably. *In electromagnetic compatibility (EMC), susceptibility is the sensitivity of a device's function to incoming electromagnetic interference Health and medicine * In epidemiology, a susceptible individual is a member of a population who is at risk of becoming infected by a disease * In microbiology, pharmacology, and medicine drug susceptibility is the ability of a microorganism to be inhibited or killed by the drug, as in antibi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hamiltonian Function
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a mechanical system with configuration space M and smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kubo Formula
The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well. General Kubo formula Consider a quantum system described by the (time independent) Hamiltonian H_0. The expectation value of a physical quantity at equilibrium temperature T, described by the operator \hat, can be evaluated as: :\left\langle\hat\right\rangle = \operatorname\,\left hat\hat\right= \sum_n \left\langle n \left, \hat \ n \right\rangle e^, where \beta=1/k_T is the thermodynamic beta, \hat_0 is density operator, given by :\hat = e^ = \sum_n , n \rangle\langle n , e^ and Z_0 = \operatorname\,\left hat\rho_0\right/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ryogo Kubo
was a Japanese mathematical physicist, best known for his works in statistical physics and non-equilibrium statistical mechanics. Work In the early 1950s, Kubo transformed research into the linear response properties of near-equilibrium condensed-matter systems, in particular the understanding of electron transport and conductivity, through the Kubo formalism, a Green's function approach to linear response theory for quantum systems. In 1977 Ryogo Kubo was awarded the Boltzmann Medal ''for his contributions to the theory of non-equilibrium statistical mechanics, and to the theory of fluctuation phenomena''. He is cited particularly for his work in the establishment of the basic relations between transport coefficients and equilibrium time correlation functions: relations with which his name is generally associated. Publications ;Books available in English *''Statistical mechanics : an advanced course with problems and solutions'' / Ryogo Kubo, in cooperation with Hiroshi Ichi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quantum Statistics
Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble (an idealization comprising the state space of possible states of a system, each labeled with a probability) that emphasizes properties of a large system as a whole at the expense of knowledge about parameters of separate particles. When an ensemble describes a system of particles with similar properties, their number is called the particle number and usually denoted by ''N''. Classical statistics In classical mechanics, all particles ( fundamental and composite particles, atoms, molecules, electrons, etc.) in the system are considered distinguishable. This means that individual particles in a system can be tracked. As a consequence, switching the positions of any pair of particles in the system leads to a different configuration of the system. Furthermore, there is no restriction on placing more than one particle in any given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quality Factor
In physics and engineering, the quality factor or factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high , while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping, so that they ring or vibrate longer. Explanation The factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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RLC Circuit
An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC. The circuit forms a harmonic oscillator for current, and resonance, resonates in a manner similar to an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping. The resistor also reduces the peak resonant frequency. Some resistance is unavoidable even if a resistor is not specifically included as a component. RLC circuits have many applications as electronic oscillator, oscillator circuits. receiver (radio), Radio receivers and television sets use them for tuner (electronics), tuning to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Resonance
Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximum amplitude response in the system. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases. All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; when there is very little damping this frequency is approximately equal to, but slightly above, the resonant frequency. When an Oscillation, oscillat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |