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| In [[statistical physics]], a '''Langevin equation''' ([[Paul Langevin]], 1908) is a [[stochastic differential equation]] describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you treasured this article so you would like to collect more info relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] generously visit our own page. |
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| == Brownian motion as a prototype ==
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| The original Langevin equation<ref>{{cite journal | title = Sur la théorie du mouvement brownien [On the Theory of Brownian Motion] | journal = C. R. Acad. Sci. (Paris) | year = 1908 | first = P. | last = Langevin | volume = 146 | pages = 530–533| id = | accessdate = 2010-08-08}}; reviewed by D. S. Lemons & A. Gythiel: ''Paul Langevin’s 1908 paper "On the Theory of Brownian Motion" [...]'', Am. J. Phys. 65, 1079 (1997), [[DOI:10.1119/1.18725]]</ref> describes [[Brownian motion]], the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,
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| :<math>m\frac{d^{2}\mathbf{x}}{dt^{2}}=-\lambda \frac{d\mathbf{x}}{dt}+\boldsymbol{\eta}\left( t\right).</math>
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| The degree of freedom of interest here is the position '''''x''''' of the particle, ''m'' denotes the particle's mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity ([[Stokes' law]]), and a [[Wiener process|''noise term'']] '''''η'''(t)'' (the name given in physical contexts to terms in stochastic differential equations which are [[stochastic process]]es) representing the effect of the collisions with the molecules of the fluid. The force '''''η'''(t)'' has a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
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| :<math>\left\langle \eta_{i}\left( t\right)\eta_{j}\left( t^{\prime}\right) \right\rangle =2\lambda k_{B}T\delta _{i,j}\delta \left(t-t^{\prime }\right) ,</math>
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| where ''k<sub>B</sub>'' is [[Boltzmann constant|Boltzmann's constant]] and ''T'' is the temperature. The [[Dirac delta|δ-function]] form of the correlations in time means that the force at a time ''t'' is assumed to be completely uncorrelated with it at any other time. This is an approximation; the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, the Langevin equation is used to describe the motion of a "macroscopic" particle at a much longer time scale, and in this limit the δ-correlation and the Langevin equation become exact.
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| Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ in the correlation function of the random force, a fact also known as [[Einstein_relation_(kinetic_theory)|Einstein relation]].
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| == Generic Langevin equation ==
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| There is a formal derivation of a generic Langevin equation from classical mechanics.<ref>{{cite book |title=Projection Operator Techniques in Nonequilibrium Statistical Mechanics |series=Springer Tracts in Modern Physics |year=1982 |first=H. |last=Grabert |volume=95 |location=Berlin |publisher=Springer-Verlag |isbn=3-540-11635-4 }}</ref> This generic equation plays a central role in the theory of critical dynamics,<ref>{{cite journal |title=Theory of dynamic critical phenomena |journal=[[Reviews of Modern Physics]] |year=1977 |first=P. C. |last=Hohenberg |first2=B. I. |last2=Halperin |volume=49 |issue=3 |pages=435–479 |doi=10.1103/RevModPhys.49.435 |bibcode = 1977RvMP...49..435H }}</ref> and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.
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| An essential condition of the derivation is a criterion dividing the degrees of freedom into the categories slow and fast. For example, local thermodynamic equilibrium in a liquid is reached within a few collision times. But it takes much longer for densities of conserved quantities like mass and energy to relax to equilibrium. Densities of conserved quantities, and in particular their long wavelength components, thus are slow variable candidates. Technically this division is realized with the [[Zwanzig projection operator]],<ref>{{cite journal |title=Memory effects in irreversible thermodynamics |journal=[[Physical Review|Phys. Rev.]] |year=1961 |first=R. |last=Zwanzig |volume=124 |issue=4 |pages=983–992 |doi=10.1103/PhysRev.124.983 |bibcode = 1961PhRv..124..983Z }}</ref> the essential tool in the derivation. The derivation is not completely rigorous because it relies on (plausible) assumptions akin to assumptions required elsewhere in basic statistical mechanics.
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| Let A={A<sub>i</sub>} denote the slow variables. The generic Langevin equation then reads
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| :<math>\frac{dA_{i}}{dt}=k_{B}T\sum\limits_{j}{\left[ {A_{i},A_{j}}\right] \frac{{d}\mathcal{H}}{{dA_{j}}}}-\sum\limits_{j}{\lambda _{i,j}\left( A\right) \frac{d\mathcal{H}}{{dA_{j}}}+}\sum\limits_{j}{\frac{d{\lambda _{i,j}\left(A\right) }}{{dA_{j}}}}+\eta _{i}\left( t\right).</math>
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| The fluctuating force '''''η'''<sub>i</sub>(t)'' obeys a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
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| :<math>\left\langle {\eta _{i}\left( t\right) \eta _{j}\left( t^{\prime }\right) }\right\rangle =2\lambda _{i,j}\left( A\right) \delta \left( t-t^{\prime}\right).</math> | |
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| This implies the [[Onsager reciprocal relations|Onsager reciprocity relation]] ''λ<sub>i,j</sub>=λ<sub>j,i</sub>'' for the damping coefficients ''λ''. The dependence ''dλ<sub>i,j</sub>/dA<sub>j</sub>'' of ''λ'' on ''A'' is negligible in most cases. | |
| The symbol <math>\mathcal{H}</math>''=-ln(p<sub>0</sub>)'' denotes the Hamiltonian of the system, where ''p<sub>0</sub>(A)'' is the equilibrium probability distribution of the variables ''A''. Finally, ''[A<sub>i</sub>, A<sub>j</sub>]'' is the projection of the [[Poisson bracket]] of the slow variables ''A<sub>i</sub>'' and ''A<sub>j</sub>'' onto the space of slow variables.
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| In the Brownian motion case one would have <math>\mathcal{H}</math>''= '''p'''<sup>2</sup>/(2mk<sub>B</sub>T)'',
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| ''A={'''p'''}'' or ''A={'''x''', '''p'''}'' and ''[x<sub>i</sub>, p<sub>j</sub>]=δ<sub>i,j</sub>''. The equation of motion d'''x'''/dt='''p'''/m for '''x''' is exact, there is no fluctuating force ''η<sub>x</sub>'' and no damping coefficient ''λ<sub>x,p</sub>''.
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| == Examples ==
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| [[File:Oscillator phase portrait.svg|right|250px|thumb|Phase portrait of a [[harmonic oscillator]] showing spreading due to the Langevin Equation.]]
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| === Harmonic oscillator in a fluid ===
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| The diagram at right shows a [[phase portrait]] of the time evolution of the momentum, ''p=mv'', vs. position, ''r'' of a harmonic oscillator. Deterministic motion would follow along the ellipsoidal trajectories which cannot cross each other without changing energy. The presence of a molecular fluid environment (represented by diffusion and damping terms) continually adds and removes kinetic energy from the system, causing an initial ensemble of stochastic oscillators (dotted circles) to spread out, eventually reaching [[Canonical ensemble|thermal equilibrium]].
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| [[File:ResistorCapacitance.png|right|250px|thumb|An electric circuit consisting of a resistor and a capacitor.]]
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| === Thermal noise in an electrical resistor ===
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| Another application is [[Johnson noise]], the electric voltage generated by thermal fluctuations in every resistor. The diagram at the right shows an electric circuit consisting of a [[Electrical resistance|resistance]] ''R'' and a [[capacitance]] ''C''. The slow variable is the voltage ''U'' between the ends of the resistor. The Hamiltonian reads <math>\mathcal{H}</math>''= E/k<sub>B</sub>T=CU<sup>2</sup>/(2k<sub>B</sub>T)'', and the Langevin equation becomes
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| :<math>\frac{dU}{dt} =-\frac{U}{RC}+\eta \left( t\right),\;\; | |
| \left\langle \eta \left( t\right) \eta \left( t^{\prime }\right)\right\rangle = \frac{2k_{B}T}{RC^{2}}\delta \left(t-t^{\prime }\right).</math>
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| This equation may be used to determine the correlation function
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| :<math>\left\langle U\left(t\right) U\left(t^{\prime }\right) \right\rangle
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| =\left( k_{B}T/C\right) \exp \left( -\left\vert t-t^{\prime }\right\vert
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| /RC\right) \approx 2Rk_{B}T\delta \left( t-t^{\prime }\right),</math>
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| which becomes a white noise (Johnson noise) when the capacitance C becomes negligibly small.
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| == Equivalent techniques ==
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| A solution of a Langevin equation for a particular realization of the fluctuating force is of no interest by itself, what is of interest are correlation functions of the slow variables after averaging over the fluctuating force. Such correlation functions also may be determined with other (equivalent) techniques.
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| === Fokker Planck equation ===
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| A [[Fokker–Planck equation]] is a deterministic equation for the time dependent probability density ''P(A,t)'' of stochastic variables ''A''. The Fokker–Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.<ref name=Ichimaru1973>{{citation|last=Ichimaru|first=S.|title=Basic Principles of Plasma Physics|publisher = Benjamin|year=1973|location=USA|edition=1st.|pages=231|isbn=0-805-38753-0}}</ref>),
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| :<math> | |
| \frac{\partial P\left(A,t\right)}{\partial t}=\sum_{i,j}\frac{\partial}{\partial A_{i}}\left(-k_{B}T\left[A_{i},A_{j}\right]\frac{\partial\mathcal{H}}{\partial A_{j}}+\lambda_{i,j}\frac{\partial\mathcal{H}}{\partial A_{j}}+\lambda_{i,j}\frac{\partial}{\partial A_{j}}\right)P\left(A,t\right).</math>
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| The equilibrium distribution ''P(A,t)'' = ''p<sub>0</sub>(A)'' = ''const×exp(-<math>\mathcal{H}</math>)'' is a stationary solution.
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| === Path integral ===
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| A [[Path integral formulation|path integral]] equivalent to a Langevin equation may be obtained from the corresponding [[Fokker–Planck equation]] or by transforming the Gaussian probability distribution ''P<sup>(η)</sup>(η)dη'' of the fluctuating force ''η'' to a probability distribution of the slow variables, schematically ''P(A)dA'' = ''P<sup>(η)</sup>(η(A))det(dη/dA)dA''.
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| The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where ''A(t+Δt)-A(t)'' depends on ''A(t)'' but not on ''A(t+Δt)''. It turns out to be convenient to introduce auxiliary ''response variables'' <math>\tilde A</math>. The path integral equivalent to the generic Langevin equation then reads
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| <ref name="Janssen1976">{{cite journal | title = Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties | journal = Z. Phys. B | year = 1976 | first = H. K. | last = Janssen | volume = 23 | pages = 377|bibcode = 1976ZPhyB..23..377J |doi = 10.1007/BF01316547 }}</ref>
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| :<math>\int P\left(A,\tilde{A}\right)dAd\tilde{A} = N\int exp\left(L\left(A,\tilde{A}\right)\right)dAd\tilde{A},</math>
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| :<math>
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| L = \int dt\sum_{i,j}\left\{ \tilde{A}_{i}\lambda_{i,j}\tilde{A}_{j}-\widetilde{A}_{i}\left[\delta_{i,j}\frac{dA_{j}}{dt}-k_{B}T\left[A_{i},A_{j}\right]\frac{d\mathcal{H}}{dA_{j}}+\lambda_{i,j}\frac{d\mathcal{H}}{dA_{j}}-\frac{d\lambda_{i,j}}{dA_{j}}\right]\right\},</math>
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| where ''N'' is a normalization factor. The path integral formulation doesn't add anything new, but it does allow for the use of tools from [[quantum field theory]]; for example perturbation and renormalization group methods (if these make sense).
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| ==See also==
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| *[[Langevin dynamics]]
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| ==References==
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| '''Notes'''
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| {{Reflist}}
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| '''Further reading'''
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| *W. T. Coffey ([[Trinity College, Dublin]], Ireland) and Yu P. Kalmykov ([[Université de Perpignan]], [[France]], ''The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering'' (Third edition), [[World Scientific Series in Contemporary Chemical Physics]] - Vol 27.
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| *Reif, F. ''Fundamentals of Statistical and Thermal Physics'', McGraw Hill New York, 1965. See section 15.5 Langevin Equation
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| *R. Friedrich, J. Peinke and Ch. Renner. ''How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market'', Phys. Rev. Lett. 84, 5224 - 5227 (2000)
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| *L.C.G. Rogers and D. Williams. ''Diffusions, Markov Processes, and Martingales'', Cambridge Mathematical Library, Cambridge University Press, Cambridge, reprint of 2nd (1994) edition, 2000.
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| {{DEFAULTSORT:Langevin Equation}}
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| [[Category:Statistical mechanics]]
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| [[Category:Equations]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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