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A consequence of describing particles as waveforms rather than points is that it is mathematically impossible to calculate with precision both the position and momentum of a particle at a given point in time. This became known as the uncertainty principle, a concept first introduced by Werner Heisenberg in 1927.
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History_of_atomic_theory
https://en.wikipedia.org/wiki/History_of_atomic_theory
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Schrödinger's wave model for hydrogen replaced Bohr's model, with its neat, clearly defined circular orbits. The modern model of the atom describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level a...
940
History_of_atomic_theory
https://en.wikipedia.org/wiki/History_of_atomic_theory
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Qualitatively the shape of the atomic orbitals of multi-electron atoms resemble the states of the hydrogen atom. The Pauli principle requires the distribution of these electrons within the atomic orbitals such that no more than two electrons are assigned to any one orbital; this requirement profoundly affects the atomi...
384
History_of_atomic_theory
https://en.wikipedia.org/wiki/History_of_atomic_theory
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In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is characteristic of a crystal. The terms " glass " and "glassy solid" are sometimes used synonymously with amorphous solid; however, these terms refer specifically to amorpho...
473
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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The term comes from the Greek a ("without"), and morphé ("shape, form").
72
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous materials have an internal structure of molecular-scale structural blocks that can be similar to the basic structural units in the crystalline phase of the same compound. Unlike in crystalline materials, however, no long-range regularity exists: amorphous materials cannot be described by the repetition of a f...
492
Amorphous_solid
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Although amorphous materials lack long range order, they exhibit localized order on small length scales. By convention, short range order extends only to the nearest neighbor shell, typically only 1-2 atomic spacings. Medium range order may extend beyond the short range order by 1-2 nm.
287
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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The freezing from liquid state to amorphous solid - glass transition - is considered one of the very important and unsolved problems of physics.
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Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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At very low temperatures (below 1-10 K), large family of amorphous solids have various similar low-temperature properties. Although there are various theoretical models, neither glass transition nor low-temperature properties of glassy solids are well understood on the fundamental physics level.
296
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous solids is an important area of condensed matter physics aiming to understand these substances at high temperatures of glass transition and at low temperatures towards absolute zero. From 1970s, low-temperature properties of amorphous solids were studied experimentally in great detail. For all of these substan...
584
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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On the phenomenological level, many of these properties were described by a collection of tunneling two-level systems. Nevertheless, the microscopic theory of these properties is still missing after more than 50 years of the research.
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Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Remarkably, a dimensionless quantity of internal friction is nearly universal in these materials. This quantity is a dimensionless ratio (up to a numerical constant) of the phonon wavelength to the phonon mean free path. Since the theory of tunneling two-level states (TLSs) does not address the origin of the density of...
561
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous materials will have some degree of short-range order at the atomic-length scale due to the nature of intermolecular chemical bonding. Furthermore, in very small crystals, short-range order encompasses a large fraction of the atoms ; nevertheless, relaxation at the surface, along with interfacial effects, dist...
598
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Due to the lack of long-range order, standard crystallographic techniques are often inadequate in determining the structure of amorphous solids. A variety of electron, X-ray, and computation-based techniques have been used to characterize amorphous materials. Multi-modal analysis is very common for amorphous materials.
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Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Unlike crystalline materials which exhibit strong Bragg diffraction, the diffraction patterns of amorphous materials are characterized by broad and diffuse peaks. As a result, detailed analysis and complementary techniques are required to extract real space structural information from the diffraction patterns of amorph...
965
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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X-ray absorption fine-structure spectroscopy is an atomic scale probe making it useful for studying materials lacking in long range order. Spectra obtained using this method provide information on the oxidation state, coordination number, and species surrounding the atom in question as well as the distances at which th...
333
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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The atomic electron tomography technique is performed in transmission electron microscopes capable of reaching sub-Angstrom resolution. A collection of 2D images taken at numerous different tilt angles is acquired from the sample in question, and then used to reconstruct a 3D image. After image acquisition, a significa...
621
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Fluctuation electron microscopy is another transmission electron microscopy based technique that is sensitive to the medium range order of amorphous materials. Structural fluctuations arising from different forms of medium range order can be detected with this method. Fluctuation electron microscopy experiments can be ...
391
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Simulation and modeling techniques are often combined with experimental methods to characterize structures of amorphous materials. Commonly used computational techniques include density functional theory, molecular dynamics, and reverse Monte Carlo.
249
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous phases are important constituents of thin films. Thin films are solid layers of a few nanometres to tens of micrometres thickness that are deposited onto a substrate. So-called structure zone models were developed to describe the microstructure of thin films as a function of the homologous temperature (T h), ...
591
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Regarding their applications, amorphous metallic layers played an important role in the discovery of superconductivity in amorphous metals made by Buckel and Hilsch. The superconductivity of amorphous metals, including amorphous metallic thin films, is now understood to be due to phonon -mediated Cooper pairing. The ro...
434
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous solids typically exhibit higher localization of heat carriers compared to crystalline, giving rise to low thermal conductivity. Products for thermal protection, such as thermal barrier coatings and insulation, rely on materials with ultralow thermal conductivity.
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Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Today, optical coatings made from TiO 2, SiO 2, Ta 2 O 5 etc. (and combinations of these) in most cases consist of amorphous phases of these compounds. Much research is carried out into thin amorphous films as a gas separating membrane layer. The technologically most important thin amorphous film is probably represente...
565
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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In the pharmaceutical industry, some amorphous drugs have been shown to offer higher bioavailability than their crystalline counterparts as a result of the higher solubility of the amorphous phase. However, certain compounds can undergo precipitation in their amorphous form in vivo, and can then decrease mutual bioavai...
354
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Amorphous materials in soil strongly influence bulk density, aggregate stability, plasticity, and water holding capacity of soils. The low bulk density and high void ratios are mostly due to glass shards and other porous minerals not becoming compacted. Andisol soils contain the highest amounts of amorphous materials.
319
Amorphous_solid
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The occurrence of amorphous phases turned out to be a phenomenon of particular interest for the studying of thin-film growth. The growth of polycrystalline films is often used and preceded by an initial amorphous layer, the thickness of which may amount to only a few nm. The most investigated example is represented by ...
867
Amorphous_solid
https://en.wikipedia.org/wiki/Amorphous_solid
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Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in unit of hertz (symbol Hz).
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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It may refer more specifically to two subcategories: Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth is equal to the upper cutoff frequency of a low-pass filter or baseband signal, ...
352
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the determinants of the capacity of a given communication channel.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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A key characteristic of bandwidth is that any band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum. For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in a POTS telephone line) or modulated to so...
467
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the ...
522
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In t...
580
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems it refers to passband bandwidth.
262
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The Rayleigh bandwidth of a simple radar pulse is defined as the inverse of its duration. For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The essential bandwidth is defined as the portion of a signal spectrum in the frequency domain which contains most of the energy of the signal.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density (in W/Hz or V /Hz) is nonzero or above a small threshold value. The threshold value is often defined relative to the maximum value, and is most commonly the 3 dB point, that is the point where the spectr...
567
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The bandwidth is also used to denote system bandwidth, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which, in the passband filter case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the ma...
630
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the stopband (s), the required attenuation in decibels is above a certain level, for example >100 dB. In a transition band the gai...
643
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak.
130
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley's law, the bandwidth refers to the frequency range within which the gain is non-...
325
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as B = 2 W , where B is the t...
793
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is often quoted relative...
472
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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There are two different measures of relative bandwidth in common use: fractional bandwidth ( B F ) and ratio bandwidth ( B R ). In the following, the absolute bandwidth is defined as follows, B = Δ Δ f = f H − − f L where f H and f L are the upper and lower frequency limits respectively of the band in question.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency ( f C ), B F = Δ Δ f f C .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The center frequency is usually defined as the arithmetic mean of the upper and lower frequencies so that, f C = f H + f L 2 and B F = 2 ( f H − − f L ) f H + f L .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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However, the center frequency is sometimes defined as the geometric mean of the upper and lower frequencies, f C = f H f L and B F = f H − − f L f H f L .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency. For narrowband applications, the...
601
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Fractional bandwidth is sometimes expressed as a percentage of the center frequency (percent bandwidth, % % B ), % % B F = 100 Δ Δ f f C .
138
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, B R = f H f L .
98
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Ratio bandwidth may be notated as B R : 1 . The relationship between ratio bandwidth and fractional bandwidth is given by, B F = 2 B R − − 1 B R + 1 and B R = 2 + B F 2 − − B F .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1. All higher ratios up to infinity are compressed into the range 100–200%.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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Ratio bandwidth is often expressed in octaves (i.e., as a frequency level) for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, log 2 ⁡ ⁡ ( B R ) .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The noise equivalent bandwidth (or equivalent noise bandwidth (enbw)) of a system of frequency response H ( f ) is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing H ( f ) when both systems are excited with a wh...
549
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The noise equivalent bandwidth B n can be calculated in the frequency domain using H ( f ) or in the time domain by exploiting the Parseval's theorem with the system impulse response h ( t ) . If H ( f ) is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then:
318
Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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B n = ∫ ∫ − − ∞ ∞ ∞ ∞ | H ( f ) | 2 d f 2 | H ( 0 ) | 2 = ∫ ∫ − − ∞ ∞ ∞ ∞ | h ( t ) | 2 d t 2 | ∫ ∫ − − ∞ ∞ ∞ ∞ h ( t ) d t | 2 .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The same expression can be applied to bandpass systems by substituting the equivalent baseband frequency response for H ( f ) .
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In photonics, the term bandwidth carries a variety of meanings:
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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A related concept is the spectral linewidth of the radiation emitted by excited atoms.
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Bandwidth_(signal_processing)
https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
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In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.
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Butterfly_effect
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The term is closely associated with the work of the mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant ...
871
Butterfly_effect
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The idea that small causes may have large effects in weather was earlier acknowledged by the French mathematician and physicist Henri Poincaré. The American mathematician and philosopher Norbert Wiener also contributed to this theory. Lorenz's work placed the concept of instability of the Earth's atmosphere onto a quan...
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Butterfly_effect
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The concept of the butterfly effect has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences.
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In The Vocation of Man (1800), Johann Gottlieb Fichte says "you could not remove a single grain of sand from its place without thereby... changing something throughout all parts of the immeasurable whole".
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Chaos theory and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the three-body problem by Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology.
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In 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.
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In 1950, Alan Turing noted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping."
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The idea that the death of one butterfly could eventually have a far-reaching ripple effect on subsequent historical events made its earliest known appearance in " A Sound of Thunder ", a 1952 short story by Ray Bradbury. "A Sound of Thunder" features time travel.
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More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published:
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"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world."
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-- The Princess of Pure Reason
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In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.
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Lorenz wrote:
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At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during whic...
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In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow (the calculations were performed on a Royal McBee LGP-30 computer). Elsewhere he stated:
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One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.
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Following proposals from colleagues, in later speeches and papers, Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap o...
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The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended ...
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The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.
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Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role. Stephen Wolfram also notes that the...
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While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied to work he published in 1969 which took the idea a step further. Lore...
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In the book entitled The Essence of Chaos published in 1993, Lorenz defined butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." This feature is the same as sensiti...
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Recurrence, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (app...
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A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz defined sensitive dependence as follows:
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The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances.
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If M is the state space for the map f t , then f t displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d (.,.) such that 0 < d ( x , y ) < δ δ and such that
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for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems.
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The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map :
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which, unlike most chaotic maps, has a closed-form solution :
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where the initial condition parameter θ θ is given by θ θ = 1 π π sin − − 1 ⁡ ⁡ ( x 0 1 / 2 ) . For rational θ θ , after a finite number of iterations x n maps into a periodic sequence. But almost all θ θ are irrational, and, for irrational θ θ , x n never repeats itself – it is non-periodic. This solution equation cle...
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The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditio...
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The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions, and the ability of a tiny perturbation to create an organized circulation at large distances, are not exactly the same. In Palmer et al., a new type of butterf...
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The third kind of butterfly effect with finite predictability, as discussed in, was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in.
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A comparison of the two kinds of butterfly effects and the third kind of butterfly effect has been documented. In recent studies, it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but si...
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The first kind of butterfly effect, known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its wings on tornado formation, ...
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According to Lighthill (1986), the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review, it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement:
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Recently, a short video has been created to present Lorenz's perspective on predictability limit.
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A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore’s Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney’s lea...
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By revealing coexisting chaotic and non-chaotic attractors within Lorenz models, Shen and his colleagues proposed a revised view that "weather possesses chaos and order", in contrast to the conventional view of "weather is chaotic". As a result, sensitive dependence on initial conditions (SDIC) does not always appear. ...
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