Right, so we I promise last time that we would talk about a multi-dimensional Langevin equation for a particle in a magnetic field and then I will talk about the properties of the Wiener process itself namely Brownian motion itself. But before that just to complete what I was saying the last time if you looked at a Brownian oscillator a particle moving on the x axis bound harmonically to the origin by this potential spring force minus half whatever omega naught squared minus omega naught squared x then we wrote down the phase space distribution we wrote down a differential equation for the phase space distribution joint distribution in both x and v the conditional density in both these variables and then I said you can solve this you get a bivariate Gaussian in x and v and if you integrate over x you would end up with the distribution in v if you integrate over v you would end up with the distribution in x. We also saw that if you have the over damped oscillator namely the one for which gamma is much much bigger than omega naught the free frequency then turns out that the position itself satisfies a Langevin type equation and you get a solution similar to the Einstein-Ohlenbeck solution except that the relaxation time is not gamma inverse but omega naught squared over gamma the whole inverse right. So it turns out that v relaxes on a time scale 1 over gamma but x for the oscillator for the Brownian for the Brownian oscillator x relaxes on a time scale which is gamma over omega naught squared we saw this from the over damped oscillator scale the fact that this is an Einstein-Ohlenbeck process and so is this an Einstein-Ohlenbeck process is responsible for the statement in many books that the Einstein-Ohlenbeck process itself is called the oscillator process what they mean is that it is the position of the harmonic oscillator when you have the over damped case okay. Now it is immediately clear from this that in the over damped case gamma over omega naught squared is much bigger than 1 over gamma which is exactly what over damping means gamma is much bigger than omega naught okay. So the velocity thermalizes if you like much more rapidly than the position does is much more sluggish but there is no long range diffusion in this problem both x and v go to equilibrium distributions which are given by the equipartition theorem if you like in statistical mechanics okay so much for this. Now for a multi-dimensional motion three-dimensional motion for example of a Langevin particle if the particle is free then you would end up with an equation which says v dot now it is a vector so let me put a wiggle underneath to show that it is a vector this is equal to minus gamma times v plus square root of gamma over m a vector valued force eta noise eta here and this is the free particle Langevin equation where each of these is a three-dimensional vector so v for example has components v j j equal to 1, 2, 3 and this guy also has components eta j running over 1, 2, 3 and it is a three-dimensional delta correlated stationary Gaussian white noise which means that this satisfies an equation of the form eta j of t equal to 0 for every j and more over eta j i of t eta j of t prime equal to a delta function i j delta of t. So the different Cartesian components of this noise are uncorrelated to each other immediately. Now once you have that information by the way we can solve once again you can solve for the Langevin equation solve for the velocity and find all its properties from those of eta or you could write down an equivalent three-dimensional Fokker-Planck equation for v and solve this equation to get a generalized Gaussian solution it will be the three-dimensional analog of the Einstein-Ohlenbeck process okay. Now we know that that process is exponentially correlated we know the velocity correlation time is gamma inverse okay so we could ask what is the correlation between different components of the velocity okay and we can do that in many ways we can solve this for v and then compute the velocity autocorrelation function but we already know from the one-dimensional example we saw that in that case v was a stationary process and v of 0 v of t was equal to k Boltzmann t over m e to the minus gamma mod t. This is what we had for a single component here now it is possible to get this equation directly get this solution directly if you know that v is a stationary process we can do this by a very cheap trick and that is to simply write the Langevin equation down so let us do that so I have v dot equal to minus gamma v plus root gamma over m e to the minus gamma t in the one-dimensional case and this is a function of t so this is v dot of t equal to v of t plus eta of t and I want to find out what is v of 0 v of t in this case so what I do is to multiply both sides by v of 0 on the left hand side and then average so v of 0 v of t v dot of t average equal to minus gamma v of 0 v of t plus root gamma over m the average value of v of 0 with eta of t and what do you think this average is what is the correlation between the velocity at time 0 and the force at t greater than equal to 0 by causality you would expect that the force at a later time does not affect the velocity the output variable at an earlier time that is causality right so this is actually 0 this quantity is 0 now you might say huh what about equal times what about at t equal to 0 so at equal times what do you think this correlation would be v of t eta of t what do you think this is going to be that is a more delicate question but again I argue the same way eta of t by from this equation here controls the acceleration so the acceleration is determined by the force not the value of the velocity this is like an initial condition for this force at equal times right so again it is completely uncorrelated to each other so that is 0 as well so causality says this quantity is 0 that is the principle we have not yet invoked okay it says that the cause cannot proceed the effect cannot proceed the cause it is not anticipatory right because of that you can make this at this equal to 0 and this is some function of t but then d over dt v of 0 v of t is precisely this quantity there is no time argument here at all or if you like put a t 0 here and a t 0 plus t here and it is independent of t 0 and the time derivative with respect to t acts only on this thing here and it is therefore equal to the time derivative of the correlation function and this is equal to minus gamma the same thing which immediately says that this should be equal to therefore v of 0 v of t should be equal to v of 0 if you like squared sorry squared inside v squared of 0 that is the initial value e to the minus gamma modulus t because you have written this equation for t greater than 0 and this now this quantity is now determined from equilibrium the fact that in stationary in equilibrium in the Maxwellian distribution this quantity is just k t over m right so this immediately tells you so even without explicitly solving the Langevin equation you can actually find what the correlation function is what do you think is going to happen there I did the same thing well I write this out for each Cartesian component and notice exactly the same property as before I multiply this for v j I write v j dot is a v j here and an eta j and I multiply by v j of 0 on the left hand side and in right solve the differential equation use the same causality argument and you get exactly the same answer for each Cartesian component right so therefore what would this be as a vector v of 0 dotted with v of t in equilibrium what would this turn out to be yeah it is k t over m times e to the minus gamma t for each of these components and then you are taking a dot product so this is equal to 3 what would this be what would be of 0 cross product v of t so now I want correlations of v 1 with v 2 v 1 at an earlier time v 2 at a later time and so on unequal components what do you think is going to happen so I write the ith component of this is epsilon i j k v j and v k out here and I do the same trick as before and again compute I can solve for this thing what do you think it will be it will turn out to be 0 because the different Cartesian components are completely uncorrelated with each other because they are driven by noises which are uncorrelated with each other this property here is a delta function here so the correlation of one component of ita with another component of ita even at the same time is 0 so this is identically 0 what would happen if I switched on a magnetic field what is going to be the equation of motion yes because now we have exactly the same Langevin equation as before but now there is a term let us say the charge of the particle is q or something like that plus q over m v cross b so there is this term plus rho gamma over m that is not affected now all kinds of correlations will happen because it is clear that this velocity dependent force the acceleration in the x direction depends on the velocity in the y direction and so on and so forth so there is correlations between the different components even though this noise is uncorrelated the Cartesian components are uncorrelated because of this term here you are going to get correlations between the different velocities and you can solve this problem for a constant magnetic field exactly the same way as before once again we know from physical arguments that the energy of the particle is not changed in a magnetic field that the Maxwellian distribution is not disturbed at all in equilibrium there is a stationary distribution that is the Maxwellian distribution all that will happen is that the diffusion coefficients in the different directions are going to be different because in the direction of the magnetic field it will be the unperturbed one but in the direction of the field there will be this cyclotron motion inhibiting long range diffusion by changing the diffusion coefficient nothing else is going to happen so I leave you to compute these quantities in this case this case here so in the presence it is a simple exercise but a very instructive one you can find out what are the cross correlations in this case this is a very special kind of force here it is velocity dependent but it is linear in v once again and this drift of course is linear in v now we know that anything that is linear in the variable you can solve the problem completely you can write the Fokker-Planck equation down and solve it you can find the green function you can solve the differential equation the Langema equation itself etc. So in exactly the same way you can extend whatever we did earlier in the absence of a magnetic field to this case here we already also saw what happens in the case of the oscillator where you have a term that is linear but that is linear in x instead of v and that did not bother us either we were able to solve it so this problem can be explicitly solved can look at it exactly in fact as the particle does cyclotron motion you with the characteristic frequency what is the characteristic frequency for this cyclotron motion in a magnetic field b there is a quantity of dimensions once you have a thing like this what is the cycle cyclotron frequency where the force is q over m v cross b right so there is a time scale here this is v dot so there is a 1 over t here times the velocity there is a velocity here already so it is clear that whatever is of inverse time scale must be q b over m right that is the cyclotron frequency it is from dimensional arguments turns out to be exactly that the numerical factor is 1 so if you go to a rotating frame of reference is rotating about the direction of the magnetic field with this frequency then it is as if you do not have this field so in that frame so you must transform from v to a variable u which takes into account that rotation and then of course you can solve them for a Planck equation much more easily or you can leave it as it is and work it out including this v so I leave this computing these quantity correlations as an exercise okay now let us get back and examine this Brownian motion itself a little more carefully we have to define what is meant by Brownian motion and by this I now do not mean the physical Brownian motion which was what was discovered first when they had these Robert Brown observed among other people he first understood what the nature of this motion was to some extent when you crush pollen grains and put it in water and then you look at it under a microscope you see this irregular jagged motions jerky motion of these particles and that is what Brown described and it was called Brownian motion okay that is physical Brownian motion we know what it is due to we know that it is due to molecular collisions and so on. But mathematical Brownian motion is an idealization okay this is defined by several in several ways and I am going to define it in one particular way today but this is exactly what the x which occurs in the diffusion equation the simple diffusion equation is supposed to describe so the process x whose probability density obeys the ordinary diffusion equation in any number of dimensions is called mathematical Brownian motion and let us go back and ask what that equal diffusion equation was so you had delta p of x t over delta t equal to d d to p over dx 2 in one dimension otherwise you got a del squared out here okay with some given constant d okay and we know this comes from the Langevin equation for a free particle in one dimension in the presence of this random force eta in the diffusion limit in the limit in which t is much much bigger than gamma inverse okay. And we also saw that this d is related to that gamma by k t over m gamma was equal to this d right now we are not worried about it we just ask where what does this say what does this equation say okay and it is useful to write down its fundamental solution of this particular equation the solution which says that p and its derivatives vanish at plus minus infinity in x and we assume also that the particle starts at the origin at t equal to 0 okay so you are really solving for the green function of this differential operator so the fundamental solution is p of x comma t the normalized solution is square root of 4 pi d t e to the minus x square over 4 d t that was the basic solution now of course you started this process at some time t prime then t gets replaced by t minus t prime and if you start at some point x naught then x gets replaced by x minus x naught now we ask what is the stochastic differential equation corresponding satisfied by x corresponding to this equation we have this correspondence between diffusion processes the Fokker-Planck like equations and stochastic differential equations which we call the Langevin equation right so by that correspondence this thing here is entirely equivalent to saying x dot equal to square root of 2d times eta of t where this is a Gaussian white noise by that I mean eta equal to 0 and eta of t eta of t prime equal to a delta function so this process x I have not distinguished here for ease of writing between the random variable itself and the values it takes which is what appears here in this thing here okay this is a matter of notation one should be a little careful but it no confusion should arise so this process x is the integral of white noise if you like because you could write x of t equal to an integral from 0 to t d t 1 eta of t 1 formally formally one can write it in this fashion for particles which start from the origin at t equal to 0 this is the formal solution to this equation is this a stationary process that is a stationary process its statistical properties are independent of the origin of time but this is not remember the velocity was a stationary process when I wrote the Langevin equation down it turned out to be a stationary random process this one is not a stationary random process here and that is easily seen because all you have to do is to look at x of t x of t prime and take its average now this average we are now taking for all those particles which start at 0 at the origin at t equal to 0 right so it is a conditional average there is no equilibrium distribution corresponding to this because this guy vanishes as t tends to infinity there is no stationary distribution at all for this process here unlike the velocity process where you got the Maxwellian as the stationary distribution now what is this going to be so let me denote by this overhead bar this conditional average that its four particles which start at t equal to 0 at x equal to 0 so what is this guy going to be equal to its equal to integral 0 to t dt 1 0 to t dt 2 t prime dt 2 and then 8 of t 1 whether I put this average or I put angular brackets here does not matter because our philosophy is that the random noise which comes from all the other particles does not is not affected by the motion of this single test particle so that acts like some kind of heat bath or reservoir and effects the particle we are looking at the tag particle but there is no effect of the tag particle on the noise itself okay all we have to do is to put this in here and then you immediately see that if you integrate over t 1 and t 2 and t 2 is up to t prime and t 1 is up to some number t the integration has support over this 45 degree line therefore it gives a nonzero contribution only as long as t 1 is runs up to t prime and after that is 0 and if it runs up to t prime you can get rid of the delta function integral delta function to get rid of the t 2 integration and then t 1 runs from 0 to t prime right and had it been the other way about had t prime in bigger than t what I have shown here is t prime less than t it would run up to the lesser of the 2 always so this immediately is equal to oh there is a 2d also 2d equal to 2d the lesser of t and t prime so that is the point it is not stationary this is not a function of t minus t prime as you would expect had this been stationary you would have ended up with t minus t prime is not true however the increments are stationary because eta itself is stationary therefore I could write this as dx if you like the increment in x is eta of t dt and then those increments are stationary because eta is stationary right or another way of saying it is if I computed x of t minus x of 0 and x of t prime minus x of 0 and took the average over that you would end up with 2t modulus of t minus t prime so this thing here is a very crucial result this is a very crucial result in fact you could say that if you have a process whose average is 0 and whose correlation auto correlation satisfies this thing here and you are given that it is a Gaussian process if you are given that then all the other properties of Brownian motion follow including the fact that it is a Markov process everything else follows so that is one way of defining mathematical Brownian motion namely it is a Gaussian process whose mean is 0 x of t average is 0 at any time t and whose auto correlation is this function here some constant times the lesser of t and t prime and this suffices to define the process and it is got remarkable properties but you already see that x is smoother in some sense than this white noise because it is the integral of white noise but it is going to be rough as we will see as it is bad enough as it stands it is very different from what the velocity was that was the Onstein-Nohlenbeck process and that had an auto correlation which was exponentially dying down okay this looks very different altogether and it is non-stationary as it stands. Now some statements about the paths the possible trajectories if you like in a typical realization of this x of t and I am not going to prove this this requires now a little more mathematical machinery which I am not going to use here but here are some facts so let us call it Brownian trail if I plotted here is t and I plotted this process x of t Brownian motion then a lot of interesting facts emerge in a typical realization it is hard to plot by the way there is no way that I can actually draw a realistic realization but if I started here and did this like this then the statement is that this x does not have any bound in any short interval however short of time you can attain arbitrarily large values of x on either side of the axis okay. The second property is that while it is continuous the curve is continuous it is not differentiable anywhere in the sense that this is a very singular object this guy is very peculiar singular object okay although we have written it in this fashion it is really a very singular object this curve is kinky on all time scales has not done does not have a derivative almost everywhere what should it be in order for it to have a derivative it is clear that you must have x of t plus epsilon minus x of t the modulus of this guy here must be of order epsilon so that if I divide over epsilon and take the limit I have a finite number right but it turns out that this is less than equal to some constant times epsilon taking epsilon to be positive to the power beta where beta is less than half the fact that you have a finite positive beta shows that the curve is continuous and it said to be holder continuous with exponent beta if this is satisfied but it turns out that beta is less than half almost everywhere but it can be made arbitrarily close to half from below almost everywhere okay. So this quantity beta tends to half from below there is a set of points where beta is exactly a half but that is a set of measure 0 on the other hand beta can become arbitrarily close to half from below so it is almost certainly not differentiable because you need beta equal to 1 for differentiability and this is less than that so that is the first property the other property the next property is that these points where it crosses the origin they form the 0 set of Brownian motion so you start from 0 and you ask when does the particle come back to 0 etc this 0 set the set of points such that x of t is equal to 0 has a fractal dimension which is half in this case that set is not countable and it is box counting or fractal dimension happens to be half this is a reflection of the fact that in this Brownian motion x squared the square of the length scales like the time to burst power that property appears over and over again you could also ask suppose you start at 0 and it moves up to the positive side rather than the negative side in a given time interval what is the fraction of the time that it spends without crossing you can ask for the distribution of that fraction okay. So let us suppose that in a time interval t what is the statistics what is the distribution of the fraction let us suppose that in a given time interval t here and this is 0 we could ask for the fraction of the time let us call the t plus over t that it spends in the positive side without crossing 0 or negative side by symmetry okay. So now this fraction here has an interesting distribution it must have a probability density function first of all and it turns out that that density function let me call it f the PDF of this random variable for a given t that is a random variable t plus this PDF f of t plus normalized PDF when you integrate this from 0 to t you must get 1 of course so this is equal to 1 over and that can be rigorously established what is that graph look like what would this look like well it becomes unbounded at both 0 as well as so t plus is sitting here yeah it looks like an inverted u of some kind with some mean value at half t over 2 okay but that is the least probable that is the least probable value you would normally expect that in a given time interval t the fraction of the time the probability density function of the fraction of the time it spends on either the positive side or in the negative side of the x axis is you know the most probable value you would think is a half but it is the least probable value okay. So it is clear that most of the time this guy is spending is this particle spending its time either on the positive side or on the negative side of the axis and yet in any finite interval of time it crosses this axis an arbitrary number of times okay so it is a very weird kind of motion the cumulative distribution function 0 to some number t plus dt plus prime f of t plus prime is the actual distribution function of this fraction and when t plus is equal to t it should be equal to 1 right this is equal to sign sign inverse t plus over t and sign was 1 is the pi over 2 so this is 2 pi that is easy to derive from here complete the square and this is called levees there is an even more exotic law which says where is this particle likely to be most of the time and it turns out that if you plot a time against the following function starts at the origin so if you plot at this function 2 t 2 dt well let me let us work in dimensionless units put d equal to 1 2 t log log 1 over t square root plot this function for sufficiently small t for t less than 1 this guy is positive and this is minus that function this guy is minus square root of the same thing and you multiply this by 1 plus epsilon you would get a curve like that and this is 1 minus epsilon square root and this function is 1 plus epsilon times this square root then the particle is almost surely going to be in this range either there or here almost surely for every t sufficiently small t okay and for sufficiently large t the 1 over t in the log is replaced by t so look at what is happening I mean we sort of know that the mean square root mean square displacement is proportional to square root of t the mean square displacement proportional to t so the root mean square is proportional to root of t so we kind of know there is a square root of t just sitting here that will be like the mean square displacement but now we are making a statement about the path itself these points this is called the law of the iterated logarithm it is Kenchin's law and its characteristic of Brownian motion so all these properties and more emerge from the very simple fact that we define mathematical Brownian motion as a Gaussian process with zero mean and with correlation which look like this x of t x of t prime correlation is the lesser of t and t prime okay now you could ask is there any relation between this process and the Onstein-Ulenbeck process because in some sense we said look the Onstein-Ulenbeck process came about has an exponential correlation and it came about when we treated the motion of this Langevam this Brownian particle a little more carefully keeping track of the velocity correlation time and so on. We also said that that process is exponentially correlated and I made a statement that the only continuous stationary one dimensional continuous stationary Gaussian Markov process is the Onstein-Ulenbeck process and it has an exponential correlation right in that statement. So in that sense that is a fundamental Gauss Markov process this is Gauss and this is Markov it is not stationary though is there a connection between this and that and the answer turns out to be very interesting it turns out that every Gaussian Markov process continuous Gaussian Markov continuous of course because it is Gaussian Markov process is some kind of Brownian motion of some kind. So let me call this Brownian motion mathematical Brownian motion this quantity dW so let us put this 2d let us get rid of that 2d and let me call W of t a process such that W dot of t is 8 of t so I do not have these constants to pull around with or if you like W of t dW of t is 8 of t dt this quantity this W of t is called a Wiener process or mathematical Brownian motion and it satisfies W of t W of t prime average equal to the lesser of t and t prime and W of t average equal to 0 it is a Gaussian process which has got 0 mean and whose autocorrelation is given by this quantity in loose terms it is the integral of white noise okay and it is a process with stationary increments because this guy is stationary and that is an increment in the process okay. Now what I was going to say was lost might train of thought yes now the statement is that every Gaussian Markov process okay it has to be Markov it has to be Gaussian is some kind of Brownian motion a Wiener process in rescaled variables so if you give me an arbitrary Gaussian Markov process let us call that process I of t in distribution as far as the probability distributions of this process are concerned this thing would be equal in distribution so let me write a d here to show that the probability distributions are the same can be written as some Wiener process in some rescaled time time some other function of t. So this is an arbitrary Gauss Markov and you can always write it as a Brownian motion in some other variable we will see an explicit illustration of it I want to connect up the velocity process the Ornstein-Nohlenbeck process to Brownian motion of this kind now you might wonder how that is going to happen after all this fellow here has a correlation which is very different it is not a function of t-t prime and yet this fellow here if it is the Ornstein-Nohlenbeck is a function of t-t prime right how does this happen well the mapping is as follows you can kind of guess what the mapping is going to be let us recall what the velocity process did so we had a V of t such that the average value of V of t for a given initial condition was equal to V0 e to the minus gamma t if you recall from the Langevin equation and we found that the correlation time so if I define delta V of t equal to V of t minus V of t bar define subtract out this average conditional average here then delta V of t delta V of t prime this quantity conditional average we discovered this guy here was equal to k Boltzmann t over m e to the minus gamma mod t minus t prime this is what we discovered right this is what we saw from the Langevin equation and it follows from either the Langevin equation directly or from the solution to the Fokker-Planck equation agree okay now how is this going to be related to that well the claim is the following delta V of t is equal in distribution apart from this constant so let us get rid of it by writing square root of k B t over m this guy is t minus t prime is t is greater than t prime and t prime minus t if t is less than t prime right so let us write this as equal to e to the minus gamma t a Wiener process of e to the twice gamma t so this scale factor outside is e to the minus gamma t and the time is getting replaced by e to the t then we apply this rule so it immediately follows this thing will immediately follow the delta V of t delta V of t prime average is k Boltzmann t over m e to the minus gamma into t plus t prime these two factors and then the smaller of e to the two gamma t e to the two gamma t prime of course this is a monotonically increasing function of t so if t is bigger than t prime it is this guy and that immediately kills this factor and makes it a plus sign so you end up with this result so this is how magically this exponents where the exponents add up it immediately once you take the exponents right e to the t instead of t or e to the two gamma t immediately this correlation looks like this but in a rescaled variable so you could say that Brownian motion is the on-synol and Mc process in logarithmic time if you like or the Wiener process in exponential time gives you the on-synol and Mc process but the statement is every Gaussian Markov process can be converted to Brownian motion so you could now ask what about the path sample paths of the velocity process itself what would those look like after all the X process had this very jagged property and I said it is a fractals or zero sets and so on which are fractals and not differentiable what would happen to the V process well the crude answer is whenever you have this kind of white noise somewhere in there you are going to see this weird property always whenever there is some driving force which is a Wiener process you are going to have which is the derivative of a Wiener process you are going to have this problem always of this very very irregular parts okay so even for the velocity process that is still true but it is really smoothed out in many other ways because of this finite correlation time and so on okay now you could ask what happens in other dimensions higher dimensions and so on so that the next thing we are going to talk about is a kind of generalization I already mentioned there is a connection between these diffusion processes and the Fokker-Planck equation here we would now like to go backwards and ask what about the Fokker-Planck equations obeyed by other random variables connected to Brownian motion for instance here is one if you have diffusion in d dimensions right so you have a probability density which satisfies the diffusion equation in d dimension so it is like a whole lot of Brownian motions in higher dimensions right what sort of properties do those trails have that is one question the other question is what kind of Fokker-Planck equation or stochastic differential equation is satisfied by other functions of these variables which are which are Brownian motion for instance in two dimensions if you had x and y you could ask what about the random variable x squared plus y squared what kind of distribution will that have what about its probability density function and so on what kind of Langevin equation would that satisfy so what we will do is to write down from the original diffusion equation we write down what the solution is or what the Fokker-Planck equation is corresponding diffusion equation is for these random variables and then use this connection between the stochastic differential equation and the master equation in reverse to write down the stochastic differential equations and see what the new information we get and that is an interesting exercise so one of them would be to ask if I call this equal to R squared what about the diffusion equation satisfied by R squared what about the Langevin equation satisfied by R squared okay how does that look what are its features and so on you could also ask what about R what about R itself R squared these are different random variables you like what about the square of a Brownian motion what about the nth power of Brownian motion in one dimension say what about x squared or x cubed or x to the n what does that look like what about e to the x what does that look like and so on we will talk about that e to the x is a very weird property it is also a kind of Brownian motion it is called geometric Brownian motion and that is the one that is used in the analysis of these financial markets so this famous black souls equation which you have which people use in fine and stochastic differential equations as applied to stock market prices is essentially geometric Brownian motion also called exponential Brownian motion so we will try to write down the stochastic equations for that and see what their properties are well some properties will become they are not so intuitively obvious for instance if you ask in two dimensions for instance if you say okay the particle is moving on this plane what kind of trail does it have it turns out that this particle also has a probability one of returning to the starting point whatever it be in other words intersecting itself arbitrarily an arbitrary number of times in fact the trail if you wait long enough will fill the plane it is space filling it has a fractal dimension of 2 almost every point is visited an infinite number of times revisited an infinite number of times that is one of the properties and you could also ask is it completely unbiased it is because it says the x and y directions these motions are not correlated to each other at all but if you write it in terms of the radial variable then you see there is going to be a bias immediately because it says that although there is no drift term when you write the Langevin equation in the Cartesian components if I now look at it at some point here at some instant of time right and ask what about the radial variable what is that going to do then you see without loss of generality we will take that point here just for ease of illustration anything that pushes it outside this circle is going to increase the radial distance anything inside is going to decrease it right now assuming that you have equal kicks in both the x and y directions in an unbiased way in arbitrary directions if you draw if you say in a given kick it is cannot go further than that although that is not true it can go arbitrarily far you immediately realize that if all these points are equally probable the measure of this set is bigger than the measure of that set so you now realize that the tendency will be to increase R rather than decrease it there is more tendency to increase R and of course as you get close to the origin if you are at the origin then any perturbation is going to increase R automatically but it is true at every point in this fashion so this means that when you write the stochastic differential equation for R in addition to the diffusion term there will be a drift term as well and it is a real effect so we need to see what that term looks like intuitively I would expect that the closer you are to the origin the greater this drift will be so I expect it will go like 1 over R maybe or something like that you will see that it indeed is so and it will also be dimension dependent clearly so for every D greater than equal to 2 this effect is going to show up and we will write this down these equations down look at it so we will do that next time


And anybody want to volunteer to be first, if you have questions, we'll start with Professor Rieger. I have just a very basic question, and I'm sure this is a question that would interest many of us. I'm intrigued by this measurement of big max per hour, which is very graphic and helps us see certain things. But in terms of a quantitative assessment, how does that relate to livability, quality of life? The ability actually to make a decent living, not just to have housing and so on. But is this a wage rate that actually is sustainable and sustains a family? Well, that's a good question. The U.S. example is one. We... Wage rates in the U.S. for people working in McDonald's range now, we know. But they're less than $10, typically. So now typically more than the... I only know this because we collected these data this summer. Typically more than the federal minimum in most states, which is seven and a quarter. That's the federal minimum. There are some states, a lot of states that have higher minimums. So if there's a higher minimum, of course, they may make more. But I would say between... The market wage is not below eight hardly anywhere and probably doesn't go much above 10 or 11. Now the question of, can you get by on that? If you work 2,000 hours and you have a family, you're below the poverty line. So it would take two people working 2,000 hours to get to that and just be barely above it. So it's extremely... I mean, the interesting... It's a very low rate of pay. But it makes you realize as low as it is, think about how low it is in the rest of the world. Yeah. The average income per head per day in the world up till the beginning of the great enrichment in some countries in the 19th and 20th centuries was $3 a day. And the average in the United States now is $130 a day, which includes these four workers. So it's really rough to get along, even in St. Peter, at $3 a day. Your appreciation of the problem depends a little bit on whether you've been to someplace like India, where... I mean, there are dimensions of the quality of life that are fine, but people are very poor. On the other hand, usually poverty is a relative factor in most of those mines. I mean, you can think of it as a subsistence level thing that would tell you how much you can get by on. But most people, I think, think of something like half the median income might be a poverty level, and certainly the minimum wage rate working at McDonald's isn't going to get to that. It's not very high. On the other hand, it's an eye-opener, I think, for people to realize how privileged we are. In some sense... Here's the way I think about it. In some sense, getting your country... Getting the country up to $8 an hour for everybody. So anybody who works at all can get at least that much, working on a regular basis. That's a challenge that the world hasn't actually managed to live up to at this point. It's a big challenge, but it's a... That's sort of the challenge of development economics, is to get... Once you grow that way, the inequality issues are not quite the same. Then once you reach the level where you have a stationary growth path, which the US is more or less on, then I think inequality issues, they materialize in people's minds more. So, building on this question, if the McDonald wage is kind of the bottom of the economy, how good of this is a representation of the economy as a whole? Because if we think that some countries, the income gap is increasing, some of them the middle class is changing, and all of these things are differentially affected in different countries, what is the limitation of this index? And what kind of other index would you like to have if you wanted a similar thing for maybe middle income and high income? Yeah, lots of... The middle income, so I think the answer to your first question is if you think of the ratio of someone at the median wage, which is maybe $18 an hour in the US, $18.20, for the average hourly earnings sake, take that to the... That's still not very high, but you take that ratio to the wage of McDonald's, so it's like two and a half to one. If the US economy grew so that the guy in the middle doubled, I think the guy at the bottom would double too. Our problem is we don't have any doubling, and it's an eye-opening to see something like China having doubling of wage rates two and a half times in seven years. I mean, if we had something like that, it would just be eye-popping, but we don't have the ability to go... We have to make new technology in order to get our productivity higher, and they can really basic disengage in getting to the production possibility frontier. But it's unclear if a new technology would help anybody on the bottom or whether it would only help people to talk? I think things tend to go up, so people worried about this in China because it's true that in China there's been a big increase in inequality, but there's no question. In fact, I talked about this work in China. It was very reassuring for some of the Communist Party folks because they wonder too whether or not the corruption and other problems is creating these phenomenal empty apartments in Miami or whatever it is. So they were relieved too. I think clearly that that economy, in some sense, when the wages at that kind of pitched level grow, it is a sign that there's real growth in the economy, and you're not in a situation where you're just taking from A to give to B. So the middle, and this is kind of a point on the distribution below the middle, it probably also would be related to a point above it. But your second question, which is really interesting one, which is how do you get, would it be possible to get a measure that had the same characteristic of allowing you to get people who are doing more or less the same thing in different places? I've thought about that. We've thought about it quite a few times. There's a few things you can think of, but most of them, the access to the data is very difficult. One of the ones I thought of once was like a Coca-Cola distributor. They make Coca-Cola and bottle it more or less the same way all over the world, whether you like it or not, by the way, all over the world. Because they're concerned about sanitary, it's safety, it's all about product safety. They're concerned about it. And so probably they have a similar kind of ideas to the way that they run. They franchise bottlers, but I think they probably have a similar way. The thing is I can't figure out how to get to the information. Forget the practical aspects. What will be your ideal product? So for example, I was thinking that maybe mattresses would be really interesting. Huge price range, very little knowledge. It's kind of basically a lot of disposable income is putting into this. And you assume that the more you have, the better the mattress will be, but it's kind of a good measure of just how much money do you have to give away without knowing much about it? Well, one of the reasons we use the Big Mac price in part is there's actually a long data set on prices of Big Macs, which goes all the way because the economists collects it. We've correlated our data with theirs. They're pretty close. And they discovered when they started to collect so that their theory is that when Big Mac prices get out of range of the exchange rate, there's going to be an appreciation or a depreciation of the exchange rate. There is truth to that. There's evidence for that. It doesn't work so well when you're looking at developing countries. And the reason is for that, because of that concave function I showed you where the real exchange rate appreciates as you come out when you're on the growth path. So that is something that doesn't get accounted for in that. But we thought of it. And then there's a whole literature, there's a wonderful paper explaining which parts and actually sort of testing the idea of the pricing that takes each component of a Big Mac. It's just in the economic journal if you're interested. It takes each component and then tries to figure out whether the prices of what the price of a Big Mac is related to the price of those components and the untrated part of course is labor. And it seems to work pretty well. So basically the part of the Big Mac, which is tradeable, it does respond to exchange rate changes, but of course the wage rate doesn't because it's a non-tradeable good. That's an interesting point that's not really, many Americans don't appreciate because we're such a high wage rate that we think of everything as more or less being tradeable. But it's not really, I mean the barber has to make as much money as the manufacturing worker. Otherwise it all become manufacturing workers. But our manufacturing workers make pretty high wages. So we don't think of it as being a big issue, but it's a giant issue when you get a haircut in India and get a fantastic haircut for almost nothing. But you know there's a longer term economic point, tradeable versus non-tradeable goods that because of arbitrage, because of competition among industries and occupations and so on, the prices of tradeable goods will eventually affect non-tradeable goods. There should be some long term relationship between them. Well that's related I think to your point which is that when you think about a tradeable goods, think of an iPhone. By the way you can get the data, people that make your iPhone make about $2 an hour and most of them have never even seen an iPhone. They don't get to see the assembled part. But you think of an iPhone, which is you can buy everywhere. And I'm not quite sure why that is. There could be some non-competitive reasons or maybe some taxes or others. But the interesting thing is it's still the case that you're not comparing something which is just a raw good. It's an iPhone. So it's like you say, I mean maybe you can get a galaxy something or other. It works just as well and it only costs half as much. So then in a way you have to be very careful. That's why when I try to use the Big Mac index, the price, I'm thinking of it, I'm trying to lay out exactly what it is and use it for comparisons across over time. But I wouldn't claim that I would compare it with the price of a mattress. I don't think you've quite responded to Professor Rieger's point though. I think you were saying, allow me to interpret for you that man does not live by Big Mac's alone. And by comparing Big Mac's or income per head for that matter, we may be reducing the human experience to bread. Is that what you added? That may be one part of the question. Another part of that question is, does it allow you to live a dignified life? That's another way of framing what you just said. And then you could add to that question the other question that we haven't touched on yet. This morning we didn't touch on it either. We talked about corruption and agents of corruption. We talked about the police. But we did not talk about large corporations who might be part of that game. And here of course we have to talk about wages at the top. And so the difference here is not merely how do we raise questions at the bottom, but how does that relate to how everybody else lives, how the rest of the population lives. If you look at the Big Mac factor in a poor country, that's one thing. If you look at it in this country, it's another question. And then you have to answer all these questions. How does dignity relate to it? How does a McDonald's worker relate to somebody who's not a McDonald's worker, a university professor, a manager of a corporation and so on in a particular social context? So bringing all of that together, the question is what does it really mean to look at those kinds of data? And my question for you is how could they be helpful for us to ask these bigger questions? Well, let me take the first part. Because I didn't respond to that really. What does it tell you about the value of your life? The value of your life, how you're living, how happy you are in some dimension? Because you certainly don't have to have material goods for everything. And there are some things, most economists don't try to measure other things like that. There are exceptions, but most people don't try to measure. I occasionally dabble in trying to measure things like that, but very difficult measurement problems are probably not something most economists try to do. There are some exceptions. A good example would be environment. So clean water, environmental pollution, these are things which clearly have both effects on the quality of your life, but also on things that are related to other aspects of your quality, directly as well as things that are related to the quality of your life indirectly, like how long you live, or whether your children have problem learning in school or something else. So those are probably economists would be interested in that, but it's much harder to say something using measurement tools like the ones I have about the broader question of, there are measures of happiness. People ask directly, how happy are you? And some people, there are many anomalies in the happiness measures. They do seem to go up with income. Up to a point. No, they go up in a concave fashion. But then there are many puzzles, like why are the Danes so much happier than the French? No, the surface of it doesn't really look obvious. But there are indirect measurements people use. Psychologists mostly and sociologists have used them. A lot of economists use them now, and you can ask whether, the thing is there are so many aspects of those measurements that are fascinating that respond to your question. For example, people that lose a limb, you would think a person who lost their limb would be very unhappy. And the evidence is that when they lose a limb, they become very unhappy, but then they recover. Their happiness, they come back to another level. So I don't know whether that means the measurement isn't really telling you how happy they are, or whether it tells you that we can all get used to everything. Maybe it does. You might use some basic measurements, like for instance education. I mean somebody who loses a limb depends on healthcare. Those kinds of things can be measured to some degree. Now what that means for the rest of your life is another question, but there are some qualifiers I think that might actually help answer that question. Well, that's true. That's absolutely true. Health is a pretty good example. So I don't know the answer to this, but it wouldn't surprise me if the wage rate at a real big max per hour was probably correlated with mortality rates. Longevitys are probably higher in countries with, I don't know, but that's something I could actually do. There is the question about longevity. I think it is true that longevity of Russian men is finally going up, whereas it had been going down for a long time. But yes, longevity, and of course it plays a big role in thinking about the economics of health because there's the question of, and it gets very sticky, because there's the question of, well, let's take a simple example in healthcare. Kidney dialysis.  I can do a kidney transplant. Kidney dialysis costs $50,000 a year or more. We all agree in the United States that you're entitled to kidney dialysis. We know that because there's an amendment to the Social Security Administration that provides it free of charge to every American, regardless of your age, anything about you. So kidney dialysis is one of those examples where we have universal healthcare. Probably we're all happier with that, but it's difficult. I couldn't put a price tag on it. I mean, how would you actually... We clearly voted to do that. We all know that we could be the one that gets the kidney dialysis. They might need it. We don't know. Anyone at this table could come to a point where they need it, but it's hard to say how you put a price tag on that. But I appreciate it. In healthcare, it's especially important because a lot of times you're thinking about, are you going to spend extra money on something? And then the answer, the question you give back is, well, so how much does that preserve life? And of course, it gets worse because then you really want to ask, well, in what quality of life? So people have things they call the quality of a life year. I have a very...  Can I interject and just have one question from our audience? We're going to have to cut this short so we can stay on schedule, but we'd like to at least honor our audience with one of their questions. In the U.S., if red and blue states are overlaid on the real wage map, red states appear to have lower real wages. Why do lower real wages seem to correlate to red voting patterns and higher wage rates to blue? Now that is a fascinating question. It's a deeper question, actually, than we can really respond to. It is a fascinating fact, it's been known for a long time, that if you look at the relationship between voting blue, let's call it blue, because I know nowadays blue can be in green or something else, but let's say the probability you vote blue, that's an increasing function of income in every state. I'm sorry, a decreasing function of income in every state. Voting red is an increasing function of income in every state, and yet if you take state averages, it's negative. So the point that was made by that questioner, which is the richer states vote blue and the poorer ones vote red, that's true, but at the same time within the blue states, the richer people vote, are more likely to vote red. So there is this deep puzzle as to how it is that the aggregation of people into states, which is so important for a political process, creates this situation where the micro-relationship is not the same as the macro one. And I don't really know the answer to it. When you say to someone, higher income people are more likely to vote red than blue, you say, oh, that's understandable, that's the Republicans and such. But then you say, but the states where they have a lot of high incomes, they actually vote blue. So what's going on? It's a mystery to me. I don't understand it either. I live in a blue state, of course, New Jersey, which is probably about as blue as they get. But I don't actually understand why it is. It's a puzzle. Well, unfortunately, we're going to have to leave that puzzle and many others to be answered hopefully over the next day and a half. So we will break now and come back to here, Professor Rieger, at 3 o'clock with some music preceding that. So thank you very much.


back to movement matters. I'm your host Christine Linders, physical therapist and board certified orthopedic clinical specialist. It's a fact. Music training is mindfulness training. I know it helped me through the worries, stresses and strains the pandemic was putting on my mind and my body. Let's look at image number one to see what music had done for me. I love this picture because it is just epitomizes the calm and the joy and the serenity that I have felt when I'm playing ukulele and I'm not even good but when I go to play that instrument that's how I feel so it doesn't matter if it's work stress, job stress, pandemic stress, pain in my body stress. I play the ukulele and that's what I feel. So today I have a very special guest who will show us how learning music can be an effective way to decrease your stress and anxiety during tough times, decrease depression and even pain, promote a healthy brain and prevent cognitive decline as we age. She not only has been teaching music for over 46 years but has a master in psychology. Let me welcome Eilisa Peralta. Thank you so much Eilisa for coming. Thanks Christine. That was great. That was great. Now sunflowers and the and the keyword there is play. You have a very playful way of going about things and that's that's great. I love it. It works. I want to know from you first how with all the stresses that everybody's been going through and not just the pandemic but just everyday little things like the toilet broke, the showers clogged, your door isn't locking and you're on your way out to work. What do you do? How can music help with some of those stresses? What are the benefits of music? I know what they are but from you. Kind of piles up. Those are daily hassles, right? Daily hassles just pile up and it can be just as bad as chronic stress. Playing music, actively playing music, not just listening but actively playing engages your whole brain.  Allows your brain to kind of reset, regroup and go at it again. Yeah, I did find that and I was telling you how I was in the midst of working full time. I'm trying to get a book out that I wrote six years ago by myself self-publishing. I'm taking care of everything. I'm worried about my family on the east coast and all of a sudden I'm finding myself playing a keyboard that I got from a patient playing my harmonica. I don't never took music lessons and now I'm taking lessons in the midst of all of this mayhem and every time I got stressed I didn't even think, oh, let me play my ukulele that you stress me. I just grabbed the thing and started playing and I, for those like 30 minutes to an hour sometimes, I would find that time in a schedule that had no time and it changed my, I feel like it changed my brain chemistry. It changed my outlook and then I went and worked on my projects and I felt like there was no worries, no pressure, nothing on my back anymore and I found an extra hour or 30 minutes in my day that I didn't even have before. Yeah, it helps you focus and you learn to focus. So the focus that you were using with your music, you kind of take that focus with you to your other project. You know, it, it's generalizes out. You've got about it really well. I'm impressed. Well, thank you. Thank you. Well, thanks to your help. So, so how did you get interested in mindfulness and, and music, I guess the connection or? Yeah, after we talked about that before, I, I sat down and said, how did I get you interested? I was a single parent trying to put myself through school, through college, and I got interested in stress. So I decided to focus on, to learn about it. So I went from stress to mind-body interactions. The influence your mind has over how your body reacts to meditation. I got into the clinical psychology program here at UH and I went to a workshop with John Kabat-Zinn on mindfulness meditation. Followed that with a year-long practice at all in the pain management program, which was based on meditation. I wrote the APA Encyclopedia of Psychology chapter on health psychology and meditation. So, and I used it. Everything I learned I used. So, yeah, I've had my stressful moments throughout my life. And it's always worked for me. You know, I was using it kind of unconsciously, not really knowing. My caregiving for my mom, you know, I took my guitar with me when I went back to Texas and I would play at the end of the day. How did you feel like before you started and then after, like after taking care of her and then at the end of the day grabbing? Like, what was that shift in your, like, relax, way more relaxed and, and, and stronger. You know, I remember one time, I think I told you about this. I was out on the patio, it's a patio in Texas that's all and I hear. But I was out there with my guitar and I was playing and I suddenly realized my sister was shaking my shoulder and saying, at least it's dark. You need to come in. And I didn't even realize that you were lost in it. I think that's, it's kind of an amazing thing, music. And for someone like, like we're so different in that, you know, music so well. And I just like playing, but I don't know the structure. It's like something I never experienced as a kid to learn notes and things. But yet I've played lots of music, because I can, you can learn other ways now without knowing the, all the little notes and things like that. But that feeling that, feeling that I was driving towards after being stressed to clear my mind. And I've done mindfulness before I took a study on like the amygdala and hippocampus and how when you're chronically stressed, you know, they did functional MRI scans of the amygdala organ looking bigger and hippocampus looking smaller and people saying, I can't remember anything. And then doing mindfulness training and then re scanning the brain and seeing how those brain organs had changed shape. That was kind of mind blowing for me as a physical therapist with pain and stress and trying to find a way to, I think like you said, get that, reset your brain or that focus on one thing, which like a mindfulness meditation does. I just never really linked it up to music until I found myself reaching for wanting craving to play something, you know, and it sounds great. It's nothing new. I mean, it's top notch musicians slip into that zone all the time. You know, it's just, but I'm thinking of why wait that long, to reap the benefits of a meditative approach to play. You should learn from the get go. Yeah. And then you get there faster going, going more slowly, more mindfully, means you're not repeating errors. So, which means you repeat errors, you get good at them. You know, that's the problem. So, yeah. Slow down. Slow down. Practice it correctly. What kind of age ranges can this mindfulness music help? Is it from kids to adults? Yep. Kids to adults.  You're making any mistakes, you're probably going too fast and you're not focused enough. So, that's your, that's your, your sign that you need to think a little deeper into that meditation. I could see that being a huge benefit for kids today with the kind of boom of devices and all the extraneous things that are bouncing off their little brains. I know one of the sources of my stresses in the technological world is texts and emails and voicemails that are out there. Whereas before it was a voicemail on the phone when you came home and now there's extra communication that when I come home at the end of the day, I'm like, oh my gosh, wait, I gotta check that and I gotta check that. And I just, I don't have the time for that, but maybe it's, I don't have the emotional time to get to that. But now with children who are getting, you know, iPads and YouTube and all this extra stimulation, I think this would be a great thing for their brains to focus on something and it's learning and it's tactile. I guess even kids who have disabilities, it would probably be a great thing for to get their brain to focus or maybe even ADD. I'm not sure. Have you had experience with that? Oh yes. I wish I'd known what I know now way back when, you know, we can all say that I guess, but I had a student years and years ago, I wasn't even in Honolulu, who had pretty severe ADHD and I was working in a small room with a drum set and shelves.  And I literally had to get him down off the wall. He was literally climbing the shelves, you know, so, so scattered and so frenetic. If I'd known then what I know now, I think I could have gotten him to focus. Yeah. I find that, I find that interesting. I have a friend who's an occupational therapist and she works with kids in the home who have different issues, ADD being one of them and autism and other things and getting them to focus on one task and helping the family to help them, helping the family communicate with them so they don't get so scattered and kind of overloaded. And I need to talk to her because I can see music as being one of those things to channel that energy that's all over the place, like bring it into one spot. And I never considered, I never thought of that until actually this moment. So do you know, do you have any examples of adults who have used music over your time other than me that has helped them to, you know, heal from an illness or get through a difficult time? Just playing cope. Yes, quite a few actually. People in high stress jobs, you know, I've seen this over and over again. One attorney told me when I'm doing this, I cannot think about anything else. It's a real break, real time out, obviously. And if you're playing instrumentals, finger picking, focusing on a move, a piece, the sound, whatever, it really does give you that time out. I had a young woman with bipolar disorder who really got herself in a lot of trouble. And she was a rapid cycler. She was up and down about every three weeks. And she told me that her guitar and the music was the only thing that stayed constant throughout the ups and the downs. It was very, very important to her. And it's a beautiful thing. That is, yeah. Yeah, those are the examples that kind of put hair on my, the goosebumps, the chicken skin. Because we all have our different things that we struggle with. And when you can find an avenue to channel that and improve yourself or just get yourself under control. And it's natural. You pick it up. You can play it. I think that's amazing to get that. I really, yeah, I would love to have more people play music for that reason. Just a joy and kids always love to mess around because it's fun because the sounds and things. And I know adults, we can get frustrated if the sounds aren't right. Do you have any tips for, you know, like, ah, I'm not getting there fast enough or the, oh, geez, I just can't seem to get this. Well, I know what your way is. But for the viewers, what's your way to help an adult get over that hump where they're, they're just like, I think I came to you and I said, I'm disgusted. I'm disgusted with myself. And I love this instrument, but I was disgusted with where I was at. And I think you said, well, you just hit a plateau. You don't stink. You know what I mean? You just hit a plateau. You know, and you gave me your little on the website, which is somewhat, maybe you can pull up the website link. The website is such a wealth of information. And you had mentioned on the website that you had your core things. And what I said about myself was not one of those. And to stay, you know, to stay encouraging myself or what do you tell people? It's the process to count. So on my website, there are a few books. I'm going to be adding to that. But books that explain that far better than I do or can. And I recommend them because they agree with me. I was, yeah, this is learning to love the process. When you say I am not there yet, or I don't sound good, you've got a goal in mind.  Am I heading in the right direction? And that's, that's its purpose. It's focusing on your very next step. Oh, and just that, you know, what I'm doing right now and what's my very next step, that's all you need. And if it's just, you know, rinse and repeat, you just keep going. Persistence is the goal. You just, you just keep going. I, when I had my stroke, which was 2015, yeah, that was a bit of a surprise. It was a mild stroke. It affected my left hand, my speech, my left leg. And I ended up in the rehab hospital where I've, you know, 17 years before I had been there, you know, interning in a mindfulness, in a mindfulness program. So mindfulness and meditation. So here I am on back in the rehab hospital as a patient. And I asked for my ukulele. At that point, I couldn't hold down one string on the uke, but I just kept kind of like tapping at it. And then I got to where I could hold it down. And then, okay, and then I tried another finger and got my stuff worked up to a scale using these three fingers. And then I managed to get my pinky into it. And then I tried a chord. I couldn't hold down both of them at the same time. That, that was a, a epiphany for me. That's, I've seen kids struggle with that. So I, and I get it now. Hold it and putting pressure on both fingers simultaneously. I got that. And then, you know, I just kept going. It's interesting to talk about the epiphanies when you lose function. I think the, what do they say, you know, walk a mile in somebody else's shoes. And I know for me as a physical therapist, and now, and for you as a music teacher, having gone through that, I've walked a mile in so many of my patients shoes well before they did. So you have that insider knowledge that, that's not just experience for you playing music or experience for me as a physical therapist with patients, but experience in myself as to, I need to be able to do this. What do I have to do? And you can't really teach that in school, right? And so you were like, wow, I can't, I can't hold that down. I know what they experienced. And then you found a way. I'm big on saying, I will find a way for people to do what they love without or with less pain so they can enjoy their lives to the fullest. And that's what you're doing, finding a way. And that's what like you've helped me and I'm sure everyone like that. Yeah, I like that. Find a way to do it. And it's different for different people. Yeah. No, but just staying focused on what you can do or what you're trying to do right now, and finding a way to do it. Yeah. And not letting it blow up. Soon as you start looking at, you know what? I was back to work in three months full time. They had given me, you know, instructions to go for outpatient and OT and speech, and all three. But when they did their initial assessment, the outpatient therapists, they just discharged it. Discharged me. They said, you don't need it. I was recovering so fast and the therapist and the neurologist attributed that. Now they're just postulating, but four therapists and three neurologists said the same thing. It's the musical instrument because you have so many more connections. It's a whole brain activity. And I really want to get this in before we run out of time. Yes, I love it. Yeah.  You use your brain in so many ways. A musician's brain actually is different. That tracks the corpus callosum that links the two hemispheres is actually thicker in musicians because you're going back and forth, back and forth, back and forth all the time. And it gives you more resources. And they told me that when the brain has to rewire after an injury, an illness, traumatic brain injury, recovery from an addiction, whatever, you've got more resources to work with. And so the brain just routes right around it. I think that's probably why I recovered so quickly. I believe that because my first job in San Diego as a new physical therapist was a cute rehab. And so I worked with strokes, brain injuries, burn patients, everything, and multiple trauma. And we always talk about making those remaking the neural connections, especially like with a severe stroke where the leg and the arm are or limp, there's nothing happening. Yeah, I never got that line wasn't that bad. I could still move. I just couldn't do anything. I didn't have any coordination, no strength. It's still the same thing though, right? Except for what we do in rehab is we're having them use a lot of their left, their noninvolve side. And I have them look at their right side because of the brain mapping. If you don't, yes, he doesn't map the connections and then that forgets and it doesn't come back. But you, even though you got mild, you're using that and looking at it. Like you said to me the last time I was there, look at your fingers. Like look at them hit the strings. And that's also mindfulness, incredible mindfulness, but also to get me to focus on I can't use this finger because I had so many stitches when I was a kid. I'm having trouble. I was 10 and I never had hand therapy. And I never had a problem until I was like, at least I can't, I can't get my pinky involved. And we were trying to work on getting that involved and looking at it helped because it's kind of had a mind of its own. All these 30 something years, maybe more than 30 years, I hate to say, but and now for the first time, I'm trying to get it to function in a specific way and looking at it helps. I had to like at first pick it up and move it there, but now I can move it there. And I had to look at it.  Eventually it wiggles. Your brain, body connection is happening over and over and over again. And I brilliant. So how can people, so how can people get in touch with you on your website? Is it best for them to email you or go to the website, the contact button? If you could pull the website up. It's alisa guitar.com. Yep. Guitar is my last name. I know. I love that. Is it is it best if they go to your website and yes. Yeah, okay. Go to my website. You can email the emails right there right top. Okay. Yeah. Good. Good.  But when you you told me something when I was in there on how you like to teach and how you watch a student to see what they need.   I like the way it sounds. And, you know, how do you how do you move me through the lessons? Like what's your approach? Well, I think a teacher's first job is to learn the student. So you kind of need to know why the students really doing it, you know. And the kind of music you like what gives you joy. So that we know which direction it goes. It gives me tools and I often will ask for a wish list.  What do you like? Do you sing?  I usually start with just accompaniment first, but occasionally there's somebody with, you know, with a music background who'd rather go straight to classical. So we work, we work it out the approach. And I need to know, you know, what you need to do for you. Yeah, that's key. Yeah. So getting getting people to learn to move into that, especially high, high achievers, high achievers is so performance oriented. I had a cardiac surgeon who was just like fanatical about that. Got to work really hard at it, got to practice and it took him a long time. He finally came in and told me, well, I got my mind around it. It's the process that counts. And I don't have to worry about how well I perform it as long as I'm, I'm take, you know, doing it correctly each time. And he made quite a quite a quite good progress. That's that I think maybe they call that that paradigm shift in your mind where you don't know. And then you finally go, I get it. It's the process. It's not that I have to perform at such a high level. It's this process of learning and growing that is so valuable. Yes. It's the flow, you know, the mind. It's mind body. It's a mind body connection. The mind, to the fingers, the ears, the brain, it all works together. It's a beautiful, beautifully orchestrated actually. Yeah, no, it's lovely. It's lovely. So we'll wrap it up so that you can get to your lesson. Thank you so much for joining me. This is a lot of fun. I think you know, this is wonderful. I just want to say to the viewers that everybody needs to go on ilisaguitar.com and look at some of the resources and contact of her lesson because it has transformed me and and my appreciation for music, but also the fun that I've had when I've played. And so thank you, Elisa, for coming and thank you everyone for joining us. Thank you for having me. I enjoyed it immensely. This is great. And remember, life is better when you listen to your physical therapist and your guitar or ukulele teacher. Aloha.