interferometric setups are at the heart of several high - precision measurement schemes , ranging from gravitational wave - detectors to laser gyroscopes and clocks synchronization protocols . in particular ,a direct detection of gravitational waves is supposed to come from laser - interferometric detectors , where they induce a measurable variation of the optical paths of the light beams traveling along the arms of an interferometric setup .inteferometers are usually employed to detect small perturbations , so that detectors must be able to measure distance changes with remarkable precision and any source of noise must be carefully removed .indeed , recent technological advances in precision lasers , vacuum technology and optical systems had made it possible to greatly reduce classical noise .however , an unavoidable constraint to the precision with which optical signals can be measured arises from the quantum nature of the electromagnetic field .the so - called standard quantum limit ( sql ) , sometimes also referred to as _ shot noise _ , results as a consequence of the uncertainty relations existing for the quantum field operators .  a way to partially circumvent the effects of quantum fluctuations and increase interferometric precisionis to exploit the use of squeezed light or other nonclassical states of light , also in the presence of inefficient detectors .the implementation of squeezing - enhancement techniques in interferometric detectors has been a great challenge : after a large number of experimental and theoretical studies it has now became possible to take advantage of squeezing so that the next generation of interferometers will be equipped with quantum technologies .  in order to effectively exploit the potential improvements offered by quantum signals , the statistical analysis of the data should be refined as well . in this context , the key point is to improve the ability of discriminating whether a signal contains instrumental noise only or also a trace of a signal .this problem , which arises in the form of a hypothesis testing problem , might be conveniently studied by means of bayesian inference . however , a bayesian approach may be challenging since it requires the full knowledge of the system under consideration in order to properly introduce a suitable prior , together with a high computational power .indeed , in a bayesian framework we have to provide explicit models of the signal induced by a perturbation to achieve , after specifying our prior knowledge , the odds ratio of the detection hypothesis over the null hypothesis , corresponding to absence of signal . in this context, it may happen that the lack of knowledge about the system increases the false alarm probability .in other words , the posterior odds may favor the alternative hypothesis , i. e. presence of perturbations , even if no signals are actually present in data .in particular , when a diffuse prior distribution is taken , a bayesian approach to hypothesis testing leads to conflicting evidences compared to a frequentist approach .such a counterintuitive situation was studied by jeffrey together with lindley , who first referred to this disagreement as a statistical paradox .the relevance of bayesian analysis in interferometry has been recognized since a long time .the interest raised for at least three reasons . on the one hand , common interferometric estimators are known to achieve optimal performances only at a specific working point , thus resulting in interferometric protocols that do not allow the measurement of arbitrary phase shifts . besides , in order to estimate small fluctuations around the working point , the interferometer has to be actively stabilized by the addition of some feedback mechanism . finally , bayesian estimators have been shown to achieve the asymptotic regime , where they saturate the cramer - rao bound , already with few measurements , thus representing a convenient choice in any setting where resources are limited or the samples involved in the interferometric setup are fragile .  in this work we address in details the occurrence of the _ lindley paradox _ in the analysis of data coming from optical interferometry .the importance of the topic seats in the fact that often , e.g. in the framework of gravitational antennas , each event is of great relevance and should be analyzed in the most refined way as the above mentioned debate concerning cosmological measurements suggests .more generally , it is often the case that besides the estimation of small fluctuations , interferometric measurements are involved in monitoring a given physical configuration , where the information coming from the experimental data is exploited to statistically discriminate between two hypothesis . here ,we first consider homodyne detection , then classical - like mach - zehnder ( mz ) interferometer , and finally we consider the case of squeezed - enhanced quantum interferometers . for the mz interferometerwe start from the analysis of the ideal case wherein the detectors are perfectly efficient , and a widespread prior is assumed , and then proceed by assessing the realistic case where inefficient detectors and a concentrated prior distribution is employed .as we will see , the lindley paradox may indeed arise in optical interferometry . herewe show how to mitigate the conflict between the bayesian and frequentist approaches by using suitable priors .the paper is organized as follows . in sec .[ s : lin ] we introduce the notation and illustrate how and when the paradox can arise . in sec .[ s : hd ] we discuss phase estimation by homodyne detection as a suitable example to illustrate the statistical analysis step by step . in sec .[ s : mzi ] we analyze in details the occurrence of the paradox in optical interferometry , whereas section [ s : outro ] closes the paper with some concluding remarks .let us consider a random variable distributed according to a gaussian distribution of unknown mean and known variance .suppose that we want to test a sharp null hypothesis , corresponding to the prediction , against the alternative hypothesis , corresponding to the diffuse prediction . in a bayesian frameworkthis is done assigning a priori a probability and a prior distribution to any hypothesis taken into account .let us denote the priors by and , with  ] .the final state , impinging on the semi - reflecting mirror , is readily obtained using the bch formulas , which yield : the distribution of the outcomes of repeated measurements of performed on radiation prepared in state is a gaussian with expectation value and variance given by :  \ , .\label{eq:7}\end{aligned}\ ] ] the above equations lead us to conclude that we shall check whether the outcomes of measurements by means of homodyne detection are consistent with the null hypothesis in order to establish whether the initial state has been perturbed or not . without loss of generality , we focus on a specific example and set , , corresponding to an input amplitude - squeezed state. we may also fix the quadrature component to be measured : for the sake of simplicity we take . in these conditions ,eqs ( [ eq:6 ] , [ eq:7 ] ) becomes \ , .\label{eq:9}\end{aligned}\ ] ]  let us now suppose that a run returns an outcome , with . in agreement with a standard sampling - theory test based on the value of shall conclude that if is several from the null hypothesis is to be rejected : we have to say that a phase shift occurred and the state has been indeed perturbed .in order to provide a bayesian assessment for we set a value for the prior probability , say , and distribute uniformly the remainder probability over the interval  ] where we expect the phase to vary . assuming to deal with rare events we take , i. e. we assume that almost only event out of is to be associated to an incoming perturbation rather than to a random error .with respect to the prior distribution for the alternative hypothesis we initially consider a situation where no information about the signal is available so that a flat prior results a proper choice . at this pointwe can compute the posterior probability given by eq .( [ eq:10 ] ) , where is to be substituted with and the probability densities take the following expressions :    in coherent light interferometry with diffuse prior . left : we show as a function of for different values of and fixed quantum efficiency .right : as a function of for different values of and fixed photon number . given the distance between the outcome and the expectation value the probability varies slowly with .,title="fig : " ]  in coherent light interferometry with diffuse prior .left : we show as a function of for different values of and fixed quantum efficiency .right : as a function of for different values of and fixed photon number . given the distance between the outcome and the expectation value the probability varies slowly with .,title="fig : " ]  the behavior of with respect to the number of photons injected and the quantum efficiency of the detectors is reported in fig .[ f3coh ] .we note that the trend is the same for different and : the presence of non unit quantum efficiency in the detection process does not change the shape of and the same goes for the light intensity . in facts ,the overall effect is to reduce the posterior probabilities , i. e. given an outcome and we have , i.e. we get a lower value of when we lower down the efficiency of the apparatus .that being said , for a given number of photons impinging on the device and efficiency of the photocounters , we can identify three different regions corresponding to different values of the outcome . in the range posterior probability is approximately , in agreement with a sampling - theory approach . in this casethe phase shift is so little that its origin is much more likely to be a random fluctuation rather than an actual perturbation .if the posterior probability is approximately : both the bayesian and the frequentist approach affirm that the phase shift has been induced by a perturbation .at last , falls from to as the value of the outcome increases within the interval .since a frequentist analysis shall provide a value of below we have that in this region we are in presence of the paradox .  on the other hand, one may find uneasy to say that an outcome four times far from the mean value is to be associated to statistical fluctuations . in facts, we can describe what is happening for as follows .our measurement of the difference photocurrent gave a random value that was manifestly inconsistent with the expected value in absence of interaction ; however , as a consequence of the prior assumed we are led to conclude that it would be even more improbable to observe that specific outcome were it the result of an interaction .that is , what makes the paradox arise is that the prior asks us to regard at every alternative possibility as an improbable coincidence .for this reason , in a real experiment an outcome in the range should be interpreted as evidence of detection .  in approaching the most realistic description ,we now discuss the occurrence of the paradox when a less diffuse prior is assumed . herewe consider a gaussian - like distribution as a prior in order to account for our belief that a perturbation results in a small phase - shift around the expected value .in particular , we consider the so - called _ wrapped normal _ distribution , defined as follows : , \label{eq:21}\ ] ] the wrapped distribution is normalized on the interval and is characterized by two parameters , and , which correspond to the expectation value and variance of a normal distribution . in recalling that the mean value of the outcomes is we impose but we let the variance of the prior free to vary in order to assess the relevance of our knowledge about the system with respect to the statistics . notethat since is inversely proportional to the ratio , which increases as the variance of the prior assumed for the alternative hypothesis decreases , we expect that the sharper is the the prior considered the higher will be the value of .for this reason we may expect the lindley paradox not to occur when the variance of is taken to be little with respect to the one of the skellam distribution , given by eq .( [ eq:17 ] ) .we report in fig .[ fig4 ] the results obtained substituting equation ( [ eq:16 ] ) with the proper expression for the prior ( [ eq:21 ] ) , so that :   in coherent light interferometry with wrapped normal prior . left : as function of and with .the coherent amplitude is so that and .an actual disagreement with respect to a frequentist approach occurs for .right : as a function of .the continuous blue line and the dot - dashed red line are obtained for and respectively .the dashed green line refers to the case and ., title="fig : " ]  in coherent light interferometry with wrapped normal prior . left : as function of and with .the coherent amplitude is so that and .an actual disagreement with respect to a frequentist approach occurs for .right : as a function of .the continuous blue line and the dot - dashed red line are obtained for and respectively .the dashed green line refers to the case and ., title="fig : " ]  at first , we see that if the variance of the prior is taken too small then the posterior probability does not evolve ( with reference to the plotted range for d ) : we get independently from the outcome . from a mathematical perspective, this is due to the fact that the ratio appearing in the expression of tends to unity as tends to a delta . in facts ,we are assuming a sharp prior centered in the mean value both for the null hypothesis and the alternative hypothesis .that is , we are somehow stating that and are equivalent as both of them , according to our prior models , lead to a measurement of to be associated to the outcome .therefore , after a trial we can not conclude anything else but that it is to be associated to a measurement of the mean value .similarly , an evaluation of the posterior probability for the alternative hypothesis yields .this last remark makes it clear that we are providing too much prior information and bayesian inference can only confirm what we already know , without the possibility of improving our knowledge about the system .conversely , if the prior distribution is diffuse , i. e. , we recover the case of flat prior examined above , where our knowledge is poor and , in turn , for the paradox is again observed .the most interesting region is .a comparison between the values of obtained employing a widespread prior and a sharper one , see the right panel of fig .[ fig4 ] , shows that the latter distribution awards less probability to the null hypothesis , as we were expected to .however , the weight of the prior is not sufficient to avoid the paradox , which still occurs if even for small values of the variance of the wrapped normal distribution .[ fig4 ] also suggests that if the interferometer has poor detectors then the odds in favor of the alternative hypothesis will be lower .nevertheless , in recalling that decreases with it is possible to see that the net effect remains unchanged .let us now consider an input state of the form , where .that is , we replace the vacuum state in mode with a squeezed - vacuum state , containing photons .the advantages gained by squeezing enhancement have been widely studied in literature . here, our aim is to assess the convenience of employing squeezed light from a statistical perspective .that is , we address the possibility of getting by the lindley paradox when the injected beam is previously squeezed .similarly to the previous case we will evaluate the posterior probability using a flat prior , at first , and then a peaked distribution . at the same time, we will pay attention to the contribution of the quantum efficiency .the difference with respect to the classical scheme seats in the distribution of the outcomes , which turns out to be difficult to be computed .however , since the expectation value and the variance may be calculated exactly we employ a gaussian distribution for the outcomes as a first - order approximation : ^ 2/2\sigma_d^2}\ , , \label{eq:23}\ ] ] where the variance is a function of the initial working point and of the measured phase shift .similarly to what happened in the previous case , takes the minimum value at , namely : \bigg\ } \nonumber\end{aligned}\ ] ] whereas the expectation value results .  in order to test the consistency of an outcome with the null hypothesis we pose and assume that the remainder probability is uniformly distributed over the interval $ ] . before computing the posterior probability we point out that the best sensitivity is achieved for and .the posterior probability that is , then , obtained by means of equation ( [ eq:10 ] ) , with equal to , leading to the behavior obtained for is very close to that observed for its classical counterpart , i.e. the dashed green line in fig .[ f3coh ] .indeed , we can identify a region where tends to unity ( for small values of ) , a region where it is approximately zero ( to the right ) and a " transition area , where the posterior probability varies between these two limits .what matters most is that we are again in presence of the lindley paradox : if we test the hypothesis concerning the detection of a perturbation by means of the bayes theorem we will end up in a misleading analysis .it is possible to see from the plot of fig .[ fig5 ] that the situation does not change in presence of non ideal photodetectors . in the same vein ,when we consider input states containing a higher number of photons and such that the optimal conditions mentioned above are still satisfied ( i. e. and ) , the posterior probability approaches zero for lower values of .however , the contribution gained from the use of higher intensity light is not sufficient to make it disappear .  at last , we take into account the most actual case with squeezed input light and a gaussian - like prior distribution wherein the statistical description .the change in the shape of has the usual effect of decreasing the value of , as it is illustrated in fig .nevertheless , we have to say that neither in this conditions we can have a consistent bayesian statistical inference .that is , the value of the posterior probability does not decrease , not even taking an appreciably peaked prior , until the outcome is more than several far from the expectation value .   in squeezed light interferometry .left : as a function of for efficiency from to and for , , .right : the posterior probability for given distance and the same values of the state parameters ( while in ).,title="fig : " ]  in squeezed light interferometry . left : as a function of for efficiency from to and for , , .right : the posterior probability for given distance and the same values of the state parameters ( while in ).,title="fig : " ]we have addressed the occurrence of the so - called lindley paradox in the analysis of data coming from homodyne detection and optical interferometry .we found that the lindley paradox may indeed occur .in particular , we have shown that the bayesian approach is somehow misleading if a scarce information about the prior distributions is available , as it happens in the evaluation of the effect of an external perturbation , and sampling - theory should be preferred .concerning mz interferometers , we have shown that lindley paradox appears both for coherent and squeezed signals and is present for any value of the quantum efficiency of the involved detectors .on the other hand , the disagreement between bayesian and frequentist approach is less pronounced for increasing noise and may be softened by using a suitable , more localized , priors .  our results , besides being of fundamental interest for interferometry , are of practical significance for quantum state reconstruction , where calibration of homodyne detectors represents a crucial step for the implementation of quantum tomography .this work has been supported by eu through the collaborative project quprocs ( grant agreement 641277 ) and by unimi through the h2020 transition grant 15 - 6 - 3008000 - 625 .consider a physical system where the quantity may be measured and assume that the outcomes from the measurement of are distributed according to a normal distribution with an unknown mean value and a known variance .we perform a measurement of and on the basis of the outcome we want to test which of the two following hypothesis is true we assume to have some prior knowledge on the system , which may be expressed in form of some _ a priori _ probabilities and for the two hypothesis .the measurement of is , in turn , intended to upgrade our knowledge of the system .given the result , bayes theorem says that for , where is the a priori probability introduced above , is the conditional distribution of the outcomes given a hypothesis , is the overall distribution of the outcomes , independently which hypothesis is actually true , and is the _ a posteriori _ probability of the hypothesis , i.e. the quantity of interest here . using the bayes theorem we may evaluate the a posteriori probabilities , e.g. is given by ^{-1 } \nonumber \\ & = \left[1 + \frac{1-p(h_0)}{p(h_0 ) } \frac{p(x|h_1)}{p(x|h_0 ) } \right]^{-1 } \ , , \label{a3}\end{aligned}\ ] ] which , upon the substitutions and , coincides with eq .( [ eq:1 ] ) .now , in order to evaluate explicitly the posterior distribution we assign some prior distribution to the mean value under the two hypothesis . according to ( [ a1 ] ) and ( [ a2 ] ) we write and assign the rest of the prior probability as a normal distribution with variance , i.e.   it is now straightforward to evaluate the conditional probabilities as which leads to ^{-1/2 }  \,e^{-(\phi-\phi_0)^2/2(\sigma^2+\tau^2)}\,.\end{aligned}\ ] ] overall , eq .( [ a3 ] ) rewrites as ^{-1}\,.\end{aligned}\ ] ] upon substituting and we arrive at eq .( [ eq:2 ] ) .99 k. s. thorne , rev .phys . * 52 * , 285 ( 1980 ) .r. x. adhikari , rev .* 86 * , 121 ( 2014 ) . v. b. braginsky , astron . lett . * 34 * , 558 ( 2008 ) .v. b. braginsky , f. ya .khalili , f. ya . , _ quantum measurement _ ( cambridge university press , cambridge , 1992 ) . c. m. caves , phys .d * 23 * , 1693 ( 1981 ) .r. s. bondurant and j. h. shapiro , phys .a * 30 * , 2548 ( 1984 ) .b. yurke , s. l. mccall and j. r. klauder , phys .rev . a * 33 * , 4033 ( 1986 ) .m. j. holland and k. burnett , phys .lett . * 71 * , 1355 ( 1993 ) .f. w. sun , b. h. liu , y. x. gong , y. f. huang , z.y .ou , g. c. guo , epl * 82 * , 24001 ( 2008 ) .m. g. a. paris , phys .a * 201 * , 132 ( 1995 ) .the ligo scientific collaboration , nature phot . * 7 * , 613 ( 2013 ) . a. a. berni , t . gehring , b. m. nielsen , v. handchen , m. g. a. paris , u.l .andersen , nature phot .* 9 * , 577 ( 2015 ) .m. g. a. paris , phys .a * 225 * , 23 ( 1997 ) .l. pezz , a. smerzi , g. khoury , j. f. hodelin , and d. bouwmeester , phys .lett . * 99 * , 223602 ( 2007 ) .l. pezz , a. smerzi , phys .lett . * 100 * , 073601 ( 2008 ) .s. olivares , m. g. a. paris , j. phys .b * 42 * , 055506 ( 2009 ) b. teklu , s olivares , m. g. a. paris , j. phys .b * 42 * , 035502 ( 2009 ) .m. g. genoni , s. olivares , m. g. a. paris , phys . rev . lett . * 106 * , 153603 ( 2011 ) .m. g. genoni , s. olivares , d. brivio , s. cialdi , d. cipriani , a. santamato , s. vezzoli , m. g. a. paris , phys . rev .a * 85 * , 043817 ( 2012 ) .s. olivares , s. cialdi , f. castelli , m. g. a. paris , phys .rev . a * 87 * , 050303(r ) ( 2013 ) .h. jeffrey , _ theory of probability _( oxford univ . press , 1948 ) .d. v. lindley , biometrika * 44 * , 187 ( 1957 ) .z. hradil , r. myska , j. perina , m. zawisky , y. hasegawa , and h. rauch phys .* 76 * , 4295 ( 1996 ) . j. p. dowling , phys .a 57 , 4736 ( 1998 ) .g. m. dariano , m. g. a. paris , r. seno , phys .a * 54 * , 4495 ( 1996 ) .d. v. lindley , biometrika * 64 * , 207 ( 1977 ) .g. shafer , j. am .stat . ass .* 77 * , 325 ( 1982 ) .h. m. wiseman , d. w. berry , s. d. bartlett , b. l. higgins , g. j. pryde , ieee j. sel .quantum electron . *15 * , 1661 ( 2009 ) .a. kumar de , d. roy , d. goswami , phys .a * 83 * , 015402 ( 2011 ) .t. reisinger , g. bracco , b. holst , new . j. phys .* 13 * , 065016 ( 2011 ) .b. t. kirby and j. d. franson , phys .rev . a * 89 * , 033861 ( 2014 ) .x. m. feng , g. r. jin , w. yang , phys .a * 90 * , 013807 ( 2014 ) .m. g. a. paris and j. rehacek ( eds . ) _ quantum state estimation _ , lect .notes phys .* 649 * , ( springer , berlin , 2004 ) .j. g. skellam , j. roy .. soc . * 109 * , 296 ( 1946 ) .

Abstract: the so - called _ lindley paradox _ is a counterintuitive statistical effect where the bayesian and frequentist approaches to hypothesis testing give radically different answers , depending on the choice of the prior distribution . in this paper we address the occurrence of the lindley paradox in optical interferometry and discuss its implications for high - precision measurements . in particular , we focus on phase estimation by mach - zehnder interferometers and show how to mitigate the conflict between the two approaches by using suitable priors .


the gp - b experiment aims at testing in earth orbit two predictions of einstein s general relativity using precision gyroscopes .this idea was independently proposed by pugh and schiff  in 1960 .both of these authors pointed out that according to the general theory of relativity , the angular momentum axis of a gyroscope in orbit about the earth will precess about a direction normal to the orbital plane due to the gravitational interaction of the spinning gyroscope with its orbital motion , and simultaneously about the direction of the earth s rotation axis due to the interaction of the spinning gyroscope with the angular momentum of the earth .the first effect is known as the geodetic effect , and the second is known as the frame - dragging effect . on a 640 km polar orbit , the gyroscope drift rate due to the orbital motion about the earth is 6.6 arcsec / yr ( 32 / yr ) , while the orbital average drift rate due to the earth s angular momentum is 0.041 arcsec / yr ( 0.20 / yr ) .gp - b uses four gyroscopes spinning in a quasi torque - free environment and placed inside a drag - free satellite .the orientation of the gyroscopes is known thanks to an on - board telescope pointing towards a distant guide star , whose orientation with respect to an extra - galactic source is known . furthermore , the satellite rolls about the telescope axis . as the orientation of the satellite with respect to an inertial reference frameis nonetheless known , reference frames linked to the satellite are qualified as inertial throughout this paper .gp - b s scientific goal can be fulfilled by measuring the orientation of the gyroscope s spin axis with respect to the satellite .the gyroscopes are superconductive , which allows tracking of the orientation of their angular velocity vectors .indeed , a spinning , superconducting body creates a magnetic dipole parallel to its spin axis , the london moment  .this magnetic dipole is then an excellent indicator of the direction of the instantaneous spin axis . since the gyroscopes are almost perfectly spherical and uniform ( ) , the spin axis direction is a very good indicator of the direction of the angular momentum .low - noise superconducting quantum interference device ( squid ) magnetometers are thus used to measure the magnetic flux through a pick - up loop placed around each gyroscope created by the london moment plus a contribution due to a magnetic field trapped in the rotor .the squid signal is proportional to the magnetic flux through the pick - up loop .the squid output is an analog signal which , on board the satellite , is split into a low frequency ( lf ) channel which contains the london moment contribution and a high frequency ( hf ) channel . both hf and lf channels pass through a hz low pass analog filter .the lf channels then passes through an additional 4 hz analog low pass filter and an additional gain stage . in this paper , we are only concerned about the hf channel , which is sampled at hz and digitized with a 16 bit adc with a range of  10 v. this adc provides a resolution higher than the signal - to - noise ratio which is near the gyroscope spin frequency .we refer to this digitized high resolution signal as the `` hf squid signal '' .  when each gyroscope transitioned below its critical temperature , the flux due to the residual magnetic field surrounding it before the transition was trapped on its surface , forming a large number of small magnetic sources .these sources are called fluxons , and can be pictured as rigidly linked to the surface of the body , as shown for instance in .a consequence is that the fluxons exactly follow the motion of the body and create modulations , on the order of a few volts , at a frequency close to the spin frequency of the gyroscope : these modulations constitute the hf signal .the hf squid signal is sent intermittently in the form of 1.86 second long stretches that we refer to as `` snapshots '' .each snapshot contains 4096 points .we also have access to this signal in the form of fast fourier transforms ( ffts ) performed at regular intervals on the squid signal by the on - board cpu .the fft is applied every 10 seconds to sets of 4096 points of raw data , and a compacted form of its output is sent to the ground .an on - board fft algorithm is applied to second - long stretches of hf squid signal and can thus resolve frequencies sec = hz apart .this frequency resolution , called a bin , is poor : the fft data is thus further processed .this additional processing is performed on the ground , and requires computing the fft at a few frequencies .the value of the fft at the first five harmonics of the signal frequency and the one corresponding to the hz calibration signal are thus telemetered .in addition , the value of the fft at the two bins adjacent to each one of these six frequencies are also sent down , as well as the zero - frequency term . for each second - long data stretch, the on - board fft algorithm thus provides 19 data points .since the high frequency data is sampled at hz , the initial data stretch contains points and so does its fft , so transmitting 19 data points is a dramatic reduction in bandwidth .additionally , raw snapshots of the squid signal are also sent down on an irregular but frequent basis .these snapshots also consist of second - long data stretches sampled at hz .an example of the time history of a portion a snapshot is provided in the upper pane figure [ fig : snapshot ] , while the fft of the entire shapshot is shown in the lower pane .( color online ) snapshot ( time - series and spectrum ) for gyroscope 1 , taken november 10 , 2004 , width=377 ]  in this paper , we describe the method that was used to determine the frequency of this hf signal , and therefore the rotor spin speed , with a resolution better than nhz .this was critical for the gp - b data analysis because it allowed the time - varying readout scale factor to be determined to 1 part in , the orientation of the spin axis with respect to the spacecraft to be determined to 3 marcsec in 1 orbit , and the gyroscopes relativistic drift rates to be determined to marcsec / yr .the time - varying scale factor is caused by the trapped flux contribution to the magnetic flux through the pick - up loop , which varies at the rotor spin spacecraft roll frequency , the rotor polhode frequency and at low frequency . with an accurate estimate of the rotor polhode and the estimate of the rotor spin speed to 30 nhz ( discussed here ) the body - fixed orientation of the gyroscope rotor with respect to the spacecraftwas determined with an accuracy of 1 deg throughout the entire science mission , lasting 1 year .this information was necessary to determine the distribution of trapped flux on the surface of the rotor and the time varying readout scale factor .  before delving into the 3 successive frequency estimation algorithms ,we introduce notations specific to the gp - b experiment in order to explain the relationship between the gyroscope motion and the squid signal .we then describe how frequency interpolation and phase differencing , a time - domain technique , were applied to the fft data to achieve a 5 frequency resolution .we then show that this result was checked using the snapshot data .variations of these first two techniques are known and their accuracy in determining monotone signals in the presence of gaussian white noise and simple systematic effects have been studied .finally , we show how the snapshot data and the result of the phase differencing were used to run the algorithm developed by the authors which allowed a 5 parts in accuracy in frequency determination .as the fluxons attached to the gyroscope move with it , hf modulations are created in the squid signal .the fluxons create a body - fixed distribution of potential : a model for this distribution can thus be written in body - fixed frame , for instance using a spherical harmonics expansion .the squid measurement however takes place in the satellite frame - which , as explained above , is considered inertial .therefore , if the appropriate set of euler rotations is applied to rotate the body - fixed frame into the inertial frame , it is possible to express the hf squid signal as a function of the coefficients of the model of the magnetic potential distribution .the vectors , and are the principal inertia axes of the gyroscope and define an orthonormal body - fixed reference frame .we call an orthonormal inertial reference frame such that is aligned with the angular momentum of the gyroscope .the angular momentum is inertially fixed as the gyroscopes are in torque - free motion .these notations are shown in figure [ fig : bfif_part3 ] .the first euler rotation from to is an azimuthal rotation by an angle .this rotation transforms into a reference frame whose axis is aligned with . a polar rotation by an angle is then applied : the new reference frame is such that its third axis is aligned with the angular momentum . the third euler angle measures the angle by which the gyroscope has spun about since a fixed time origin : a rotation of angle about is thus also needed to obtain the inertial reference frame .consequently , the frequency at which any fluxon passes through the pick up loop is , where is the polhode frequency , such that .as the modulations in the hf squid signal are due to the motion of the fluxons with respect to the pick - up loop , the frequency of this signal is thus given by :    an expression for can be obtained from , formulas 89.4 and 90.4 :  ,\ ] ]  where the characteristic , , and the elliptic modulus , , only depend on the moments of inertia , and , the rescaled time, , depends on the three moments of inertia and the angular momentum , ( see ) and is the jacobi elliptic integral referred to as `` sinus amplitude '' and defined by : the first term in eq .( [ eqn : first_dotphis ] ) is the larger by a factor of , and the second term is a modulation at twice the polhode frequency .the three euler rotations are summarized in the definition of the spin axis : the spin frequency is thus    as and can be as large as which is on the order of mhz on all gyroscopes , lies within mhz of the spin frequency and is thus on the order of 100 hz .the hf squid signal frequency is thus close to the spin frequency .the first two steps of the frequency determination method presented in this paper are applied to the hf squid signal in order to determine .the third step of the frequency determination method presented in this paper thus aims at determining with a nhz resolution . in , a procedureis given to measure the polhode frequency with a nhz accuracy .note that this represents an accuracy of 1 part in as the polhode frequency is on the order of 0.1 mhz .therefore , from ( [ eqn : fz ] ) , an absolute accuracy better than 30 nhz in the the determination of the frequency of the hf squid signal can be achieved . in relative terms , the claimed accuracy of the frequency determination is thus on the order of 5 part in .by interpolating the fft data , it is possible to go beyond the hz frequency resolution and to determine the frequency with a mhz precision .this method is well known and has been used for several decades .it consists of approximating the signal by a pure sine around its main frequency , and finding the frequency of such a sine if it had the same amplitude diagram as the fft of the signal .we now give a mathematical description of this method .  as the interpolation is applied to the fft data which is itself computed using short stretches of squid signal , we neglect the polhode harmonics in this description .we show , without loss of generality , how frequency interpolation is implemented for a single frequency signal .let be our signal , so that its discrete fourier transform is : }   { \sin [ 2\pi(f - f_z)\delta t / 2]}\right ] \nonumber\\  & & + \frac{a_s}{2 } e^{-i\delta\phi } e^{-i 2\pi(f + f_z )   \delta t \frac{n - 1}{2 } }   \left [ \frac{\sin\left [ 2\pi(f + f_z)\delta t n /    2\right]}{\sin\left [ 2\pi(f + f_z)\deltat / 2\right]}\right ]   \label{eqn : full_fourier_cos}\end{aligned}\ ] ]  where : is the time between two consecutive samples ( sec for a hz sampling rate ) , is the number of points in the fft , n is then the total snapshot duration , is the bin frequency , is the frequency at which the fft is computed , is the signal amplitude and is the phase shift .  nearthe hf signal s frequency , , so the first term in ( [ eqn : full_fourier_cos ] ) is dominant and we neglect the second one .since the fft is discrete , we only obtain its values at multiples of the bin frequency .furthermore , as mentioned above , we have at our disposal the value of the fft at the central bin and at the two adjacent bins .let s define the integer such that is the multiple of closest to the signal frequency .we note :   as implied by eq .[ eqn : x_n ] , the term here is at most , since near the main peak the signal frequency , , is at worst one frequency bin away from . from these relations , we derive two formulas for the frequency of the hf squid signal , depending on the sign of . for :    averaging the two formulas for we and obtain : + \mathcal{o } \left(10^{-3}\right)\ ] ] for :      averaging these two formulas gives : + \mathcal{o } \left(10^{-3}\right)\ ] ]  therefore , the frequency interpolation yields a determination of the hf signal s frequency , whose error is , or hz .indeed , the frequency computed by this method typically showed a  spread , as shown on figure [ fig : interp_output ] .  the result derived in ( [ eqn : omeg_interp ] )was obtained for the simple case of a signal with only one harmonic .however , the hf signal contains many harmonics of its main frequency , and as explained in , the odd harmonics have a significantly larger amplitude .the interpolation procedure is therefore separately applied to the fundamental frequency , , as well as to the and harmonics of the fft data .the results from each of those 3 interpolations are then averaged in order to obtain a determination of the signal frequency .results for a 6-hour long data stretch for gyroscope 1 are shown in figure [ fig : interp_output ] .( color online ) frequency from interpolation , gyroscope 1 , aug .19 2004 , width=321 ]  the formula ( [ eqn : omeg_interp ] ) for the signal s frequency is valid when no window is applied .applying windows to a signal consists of multiplying it by a carefully chosen function -for example , a sine square- so the frequency components away from the main frequency are attenuated . in the case of the windowed signal ,the analytical expression of is more complex but the principle of the derivation remains exactly the same .the interpolation method relies on the use of the amplitude of each independent fft and , in practice , yields a mhz or better frequency resolution .we now present an additional method called phase differencing that uses the phase of the peak fft bin , which is the one closest to the frequency of interest , at two different times .this method yields more than two additional orders of magnitude in frequency resolution .phase differencing consists of using a precise measurement of the change in the phase of the fft computed on two distinct 1.86 second - long data stretches . knowing the time elapsed between the two data stretches, we can determine the signal frequency .however , to resolve the ambiguity , an initial estimate of this frequency is necessary .indeed , let be the phase of the fft obtained at time and the phase of the fft obtained at time .we can assume that the frequency of the signal is constant if and are close ( we indeed show later that the characteristic time of the decay in is 7,000 - 25,700 years , depending on the gyroscope ) .we then have :    where is an integer such that is between 0 and 2 .we then need to know the value of in order to extract the frequency from the previous equation .this requires knowing an estimate of the signal frequency so that : .\ ] ] the value of then has to be chosen to ensure condition ( [ eqn : f_est_cond ] ) .in order for to be known , must lie within hz of the real value since sec .this resolution is easily achieved by the interpolation algorithm .furthermore , this hz requirement is more stringent than the hz precision of the fft : the result of the interpolation is thus needed to carry out the phase differencing .then , once is computed , equation ( [ eqn : f_est ] ) can trivially be solved for the signal frequency .this new estimate has a very good precision .indeed , the phase is known with a conservative error of 10 arcsec ( rad ) , which at the gyroscope spin speed corresponds to a timing error of approximately sec , which may be compared to the short - term on - board clock accuracy of approximately sec .the main error term is thus the phase measurement , and for a sec interval , we obtain a precision better than 5 .figure [ fig : phs_diff_output ] shows the extra resolution that can be achieved with phase differencing .it should be compared with figure [ fig : interp_output ] , which only shows the output of the interpolation on the same data stretch .( color online ) frequency from phase differencing , gyroscope 1 , aug .19 2004 , width=321 ]  the large modulations in are at polhode frequency and are a systematic error in the estimation method .indeed , we used a simple model for the hf signal which only takes into account the harmonics of , whereas harmonics of the polhode frequency also contribute to the signal s spectrum .these frequencies then leak into our current measurement of . the model introduced in the last section of this article takes into account all the frequency components of the hf signal , and yields an estimate of free of large variations at polhode frequency .  as a conclusion , by interpolating each fft and using phase differencing between ffts, we can obtain an estimate of the signal frequency with an error of a few .we have thus gained 5 orders of magnitude as compared to the fft bin width of hz .furthermore , this determination only relies on estimates of the fft for a small number of frequencies being transmitted from the spacecraft .it then has the double advantage of requiring little bandwidth and using less on - board cpu time , since the interpolation and the phase differencing are carried out on the ground .typical values for obtained after phase differencing are given in table [ tbl : spin_speed ] .they have been measured on feb . 2005 at 07:39 gmt ..typical values for signal frequency obtained from phase differencing [ cols="^,^",options="header " , ]this paper describes three successive algorithms for estimating the frequency of a digital signal .the first two of them , interpolated fft and phase differencing , known previously , were properly adjusted for the needs of the analysis of the hf signal obtained in the gravity probe b experiment .the third method is new ; it is based on the accuracy achieved by the previous two approaches , and involves a nonlinear estimation of a slowly changing frequency when many harmonics of another time - varying frequency are present .the last method requires accurately modeling the rigid body motion of a nearly torque - free gyroscope , and utilizes the specially modified nelder mead simplex method .these algorithms are applied to a particular problem of precisely determining the spin frequency of the ultra - precise gp - b gyroscopes .the measured hf squid signal was used for this analysis whose signal to noise ratio was .we achieved the ultimate relative accuracy of 5 parts in , corresponding to the 30 nhz absolute accuracy for a gyro spinning at 60 - 80 hz .this accuracy level was necessary for the estimation of trapped magnetic flux on the surface of each gyroscope rotor to 1% , the resulting time variations in the gyro readout scale factor to , and ultimately the relativistic gyro drift rate to 20 marcsec / yr .

Abstract: in this paper , we present a method to measure the frequency and the frequency change rate of a digital signal . this method consists of three consecutive algorithms : frequency interpolation , phase differencing , and a third algorithm specifically designed and tested by the authors . the succession of these three algorithms allowed a 5 parts in resolution in frequency determination . the algorithm developed by the authors can be applied to a sampled scalar signal such that a model linking the harmonics of its main frequency to the underlying physical phenomenon is available . this method was developed in the framework of the gravity probe b ( gp - b ) mission . it was applied to the high frequency ( hf ) component of gp - b s superconducting quantum interference device ( squid ) signal , whose main frequency is close to the spin frequency of the gyroscopes used in the experiment . a nhz resolution in signal frequency and a phz / sec resolution in its decay rate were achieved out of a succession of second - long stretches of signal sampled at hz . this paper describes the underlying theory of the frequency measurement method as well as its application to gp - b s hf science signal .


the standard distributions in equilibrium statistical mechanics can be derived in an appealingly straightforward fashion using the principle of maximum entropy or maxent .the procedure seems first to have been employed by gibbs , and was vigorously championed by jaynes as an example of logical inference , namely the optimal determination of a statistical description of an imperfectly specified system .the essential idea is that a system can possess an informational entropy that measures the uncertainty of an observer s perception , expressed through a probability distribution over all the configurations ( microstates ) that the system could adopt . in order to represent the situation in as objectively neutral a fashion as possible ,so the argument goes , we should select the distribution that has the greatest informational entropy , while enforcing consistency with any known properties of the system , represented as expectation values over the distribution . for various mathematical and physical reasons , the informational entropy of a probability distribution over a set of microstates labelled is written and if the constraints take the form of fixed expectation values of a set of microstate - dependent quantities , then it can be shown by way of the method of lagrange multipliers that the statistical representation that makes no unwarranted further assumptions about the system is  ] not appear in the exponent of the canonical distribution in addition to ?an argument is often made that constraints are placed upon dynamically conserved quantities , which would exclude arbitrary functions of energy , but this strongly suggests that dynamics must underpin the procedure .it is our aim here to demonstrate that the framework of stochastic thermodynamics can provide the underlying dynamics in the derivation of the maxent principle .  in section[ sec : stochastic - dynamics - and ] we give a brief overview of stochastic thermodynamics , discussing the way in which ( stochastic ) entropy production is both a reflection of the mechanical irreversibility of the underlying stochastic dynamics and the basis of a measure of the change in microstate - level uncertainty of a system with time .we describe how the total entropy production may usefully be separated into three components , each with a specific character . in section[ sec : dynamical - maxent - selection ] we discuss the dynamics of mean total entropy production for a system subject to a conservative force field and coupled to an isothermal reservoir , arguing that this viewpoint provides a route to the canonical solution to the stochastic dynamics , and showing how this maps onto the traditional maxent variational principle .we see how the constraints on the variational procedure emerge as a result of the dynamical coupling of the system to the reservoir .only one of the three components of entropy production is non - zero in this case .  in section [ sec : nonisothermal ]we consider a nonequilibrium situation where a system is exposed to a background temperature gradient and show that the stationary state may be selected by the maximisation of the mean relaxational or nonadiabatic entropy production , one of the three components , or equivalently by requiring the increment in this component to be zero for all possible dynamical scenarios .we note that the average of the remaining two components accounts for a recently discussed anomaly in entropy production between over- and underdamped treatments . supporting analysisis provided in appendix [ sec : components - of - stochastic ] and a system driven by a nonconservative force under isothermal conditions is discussed in appendix [ sec : particle - in - a ] .we give our conclusions in section [ sec : conclusions ] .stochastic thermodynamics is based on a set of stochastic differential equations ( sdes ) that describe the evolution of system dynamical variables , together with a definition of the entropy production associated with a possible realisation of the motion .we focus our discussion on the motion in one spatial dimension of a single particle coupled to a set of heat reservoirs , each corresponding to a given position of the particle .the motion is described by the following it-rules sdes : where and are the particle position and velocity , respectively , is time , is the friction coefficient , is a spatially dependent force field acting on the particle , assumed for the moment to be related to a potential , is the particle mass , is a spatially dependent reservoir temperature and is an increment in a wiener process .such a starting point for discussing the stochastic behaviour of a particle in a temperature gradient is often employed , though alternatives can also be imagined .the simplicity of eqs .( [ eq:1 ] ) and ( [ eq:1a ] ) is convenient for our purpose .  given the dynamics , entropy production is defined in the fashion proposed by seifert .it is fundamentally a measure of the probabilistic mechanical irreversibility of the motion . for a given time interval , the dynamics can generate a trajectory ( where represents a function in the specified time interval ) according to a probability density function  ] that an antitrajectory starting at and ending at is generated , with the superscript r reminding us that the potential and reservoir temperature evolve with time in a reverse fashion with respect to the period .the antitrajectory is the ` time - reversed ' partner of  .the total entropy production associated with the trajectory is then defined by =\ln\left[\frac{{\cal p}[\vec{\boldsymbol{x}},\vec{\boldsymbol{v}}]}{{\cal p}^{{\rm r}}[\vec{\boldsymbol{x}}^{\dagger},\vec{\boldsymbol{v}}^{\dagger}]}\right],\label{eq:2}\ ] ] and after multiplication by boltzmann s constant and averaging over all trajectories , this corresponds to the production of thermodynamic entropy in the process . in a condition of thermal equilibrium , when the dynamics would be expected to generate a trajectory and its time - reversed partner with equal likelihood , the entropy production associated with _all _ feasible trajectories will vanish .the entropy production along a trajectory evolves stochastically , and for the system dynamics considered here an increment in is specified by the it-rules sde -\frac{1}{kt_{r}(x)}d\left(\frac{mv^{2}}{2}\right)+\frac{f(x)}{kt_{r}(x)}dx.\label{eq:3}\ ] ] the origin of this expression is described in appendix [ sec : components - of - stochastic ] and elsewhere .the second term is the negative increment in the kinetic energy of the particle over the time interval , and the third term is the negative increment in its potential energy , both divided by the local reservoir temperature .together , they represent a positive increment in the energy of the local reservoir ( which we may regard as a heat transfer to that reservoir ) divided by the local temperature .this would then correspond to a clausius - type incremental change in the entropy of the local reservoir in the interval of time .it is then natural to regard the first term in eq .( [ eq:3 ] ) as the change in the entropy of the system ( the particle ) over this period .seifert defined a stochastic system entropy in terms of the evolving phase space probability density function generated by the stochastic dynamics , such that we can write   since velocity evolves in eq .( [ eq:1a ] ) under the direct influence of a stochastic force , the rules of stochastic calculus apply when we manipulate increments of a function of . since we employ it rules it would be incorrect to proceed from eq .( [ eq:3 ] ) by writing which only applies under stratonovich rules .an additional term proportional to would appear . taking this properly into accountremoves an apparent anomaly in discussed in .the evaluation of for a specific realisation of the motion requires us to determine the evolution of the pdf as well as the system variables and : we need to solve the fokker - planck equation  that corresponds to eqs .( [ eq:1 ] ) and ( [ eq:1a ] ) , where is the irreversible probability current , with .  in spite of fluctuations in the total entropy productionas the particle follows a trajectory , it can be shown that the average of this quantity is non - negative , a property that arises from an integral fluctuation relation .this is regarded as the second law of thermodynamics in this framework , expressed as where the angled brackets denote an expectation over the pdfs of system coordinates at the beginning and end of the incremental time period .the entropy production can be separated into a specific set of components , each with a particular character .initial developments in this direction were provided by van den broeck and esposito and extended by spinney and ford , working within a framework suggested by oono and paniconi .the total entropy production may be written as three terms  with the and components defined in terms of ratios of probabilities that specific trajectories are taken by the system .we give particular attention to the first component , given by =\ln\left[\frac{{\cal p}[\vec{\boldsymbol{x}},\vec{\boldsymbol{v}}]}{{\cal p}^{{\rm ad , r}}[\vec{\boldsymbol{x}}^{r},\vec{\boldsymbol{v}}^{r}]}\right],\label{eq:5b}\ ] ] where represents a reversal of the system trajectory _ without _ the inversion of velocity coordinates , and the superscript ` ad ' indicates that ` adjoint ' dynamical rules are employed to work out the probability of its generation .our key point is that a dynamical underpinning of the maxent principle can be obtained by considering the properties of the component .the approach of the pdf under the dynamics towards stationarity is equivalent to a variational principle for its selection expressed in terms of .such a principle holds irrespective of whether the stationary state is one of equilibrium , in which case and are both zero during the evolution , or nonequilibrium such that and are in general non - zero . in both cases vanishes asymptotically and reaches a ceiling .let us substantiate these claims .it was shown in that the average values of are related to the transient and stationary system pdfs ( and , respectively ) as follows : \nonumber \\  & = & \int dxdv\;\frac{p}{d_{v}}\left(\frac{j_{v}^{{\rm ir}}}{p}-\frac{j_{v}^{{\rm ir , st}}}{p_{{\rm st}}}\right)^{2}\ge0,\label{s1av2}\\ \frac{d\langle\delta s_{2}\rangle}{dt } & = & \int dxdv\;\frac{p}{d_{v}}\left[\frac{j_{v}^{{\rm ir , st}}(x ,- v)}{p_{{\rm st}}(x ,- v)}\right]^{2}\ge0,\label{eq : s2av}\\ \frac{d\langle\delta s_{3}\rangle}{dt } & = & -\int dxdv\;\frac{\partial p}{\partial t}\ln\left[\frac{p_{{\rm st}}(x , v)}{p_{{\rm st}}(x ,- v)}\right],\label{eq : s3av}\end{aligned}\ ] ] where , and is the irreversible probability current in the stationary state .the positivity of the rate of change of and is here explicit , but is also a consequence of integral fluctuation relations for these two components of entropy production .furthermore , the unaveraged increments and take the form +d[\ln p_{{\rm st}}(x , v)],\label{eq : ds1}\ ] ] and +d[\ln p_{{\rm st}}(x ,- v)],\label{eq : ds3}\ ] ] making clear the conditions for which they vanish ( and =p_{{\rm st}}(x ,- v) ] .the sdes that govern the evolution of for a general markovian dynamical framework are given in appendix [ sec : components - of - stochastic ] .the three contributions to the total entropy production can be interpreted as follows . is the principal relaxational entropy production associated with the approach of a system towards a stationary state .once a system is in a stationary state , with , no further increments in take place .this component of entropy production , averaged over all possible realisations of the motion initiated at , namely , increases monotonically towards a positive constant , since eq .( [ s1av2 ] ) indicates that as .the contribution was denoted the nonadiabatic entropy production by esposito and van den broeck and its properties were given particular attention by hatano and sasa . is also associated with relaxation towards the stationary state , but in contrast to , its average value does not necessarily evolve monotonically with time ; there is no definite sign attached to in eq .( [ eq : s3av ] ) . if the stationary pdf is velocity symmetric , however , eq .( [ eq : ds3 ] ) shows that is identically zero . since a velocity asymmetric stationary pdfis typically associated with a non - zero mean flux of some kind , and also with an underlying breakage of the principle of detailed balance in the stochastic dynamics , this component arises in situations where there is a nonequilibrium stationary state .it was designated the transient housekeeping entropy production by spinney and ford . is also associated with a nonequilibrium stationary state , since a non - zero rate of change of its average over all possible realisations of the dynamics requires there to be a non - zero current in the stationary state .the rate of change of the average is non - negative , but is non - zero in a nonequilibrium stationary state , in contrast to the rate of change of the average which would then be zero .this is the most distinctive difference between these two components of entropy production .esposito and van den broeck referred to as the adiabatic entropy production ( and they considered it only in the context of the dynamics of even coordinates such as position ) , and spinney and ford , who considered odd coordinates such as velocity as well , denoted it the generalised housekeeping entropy production .other separations of total entropy production into three components are possible , but the choice employed here has the advantage that the component vanishes in the absence of velocity variables in the dynamics , and on average its rate of change in a stationary state is also zero .as a system approaches stationarity , all three kinds of entropy production take place , but in the stationary state only and can potentially receive increments .the mean total entropy production rate in the stationary state is represented by alone .if detailed balance holds , both and would be zero .our proposal is that the monotonic increase in towards a ceiling , or equivalently the vanishing of in the stationary state , underpins maxent for equilibrium situations , and offers an extension of the procedure to nonequilibrium circumstances .we now explore the implications of this viewpoint for isothermal and then nonisothermal situations .if the reservoirs were isothermal ( ) , then irrespective of the initial conditions , the system under consideration should relax under the stochastic dynamics to an equilibrium state with zero irreversible probability current and canonical statistics .let us examine how this works out dynamically and thermodynamically .we presume that the contributions and are zero throughout as a consequence of the associated condition of detailed balance in the dynamics , as suggested by the velocity symmetry of the expected canonical pdf .this can be checked later .if the change in total entropy production is indeed entirely given by then the total entropy production satisfies not only the incremental representation in terms of system and reservoir contributions in eqs .( [ eq:3 ] ) and ( [ eq:4 ] ) , but also the mean behaviour represented by eq .( [ s1av2 ] ) .we are therefore in a position to write the mean total change in entropy of system plus reservoir will increase monotonically until it reaches a ceiling .the equilibrium state will be achieved when becomes time - independent .we argue that eqs .( [ eq:6 ] ) are equivalent to the maximisation of a constrained gibbs system entropy over the range of possible pdfs , namely a maxent procedure .the maximisation would have its origin in the dynamics , and would not arise merely from considerations of logical inference .  to support this viewpoint , we note that the incremental transfer of energy to the reservoir , that specifies the entropy production in eq .( [ eq:4 ] ) , is equal to , the negative of the increment in the change ( with respect to ) in the energy of the system as it evolves over the time interval . since , we can rewrite eq .( [ eq:4 ] ) as next , we recognise that since is a function of system coordinates at a specified time rather than a function of a trajectory of coordinates , the average over realisations of the dynamics is a difference in the expectation of between the final and initial states .that is where the expectations( indicated by overbars ) are averages over the system pdf ; thus =s_{g}(t) ] where is a local canonical distribution , and .note that if we have a canonical distribution over velocity and vanishes as claimed in the previous section .we also write +d\ln[1+\psi(x ,- v)],\label{eq:702}\ ] ] and using we have ,\label{eq:703}\end{aligned}\ ] ] and this reduces to }{\partial v}\right)^{2}dt\nonumber \\  &  & -\frac{\partial\ln[1+\psi(x ,- v)]}{\partial x}dx\nonumber \\  &  & -\frac{\partial\ln[1+\psi(x ,-v)]}{\partial v}\left(-\gamma v+\frac{f}{m}\right)dt\nonumber \\  &  & -\frac{\partial^{2}\ln[1+\psi(x ,- v)]}{\partial v^{2}}\frac{kt_{r}\gamma}{m}dt.\label{eq:704}\end{aligned}\ ] ] where .setting corresponds to a general condition for the structure of and hence the pdf in the stationary state . for illustration , however , we proceed with an assumption that is small and inversely proportional to , which is the commonly used perturbative chapman - enskog representation , a well - established approach to solving problems in kinetic theory to first order in inverse friction coefficient .this is not the same as making an assumption of overdamped dynamics , which would involve a different specification of the underlying stochastic dynamics .we shall consider such an approach in section [ sub : overdamped - dynamics ] .we identify the leading contributions to , namely those of order and write for the right hand side to vanish term by term we deduce that a polynomial representation of can only contain linear and cubic terms in .we write and require contributions to proportional to to vanish by demanding that so that .we consider a situation where the particle is spatially confined by the potential , in the sense that the pdf vanishes as .this implies the physical requirement that the mean velocity at a given is zero in the stationary state , equivalent to the condition .this means that and hence to lowest order in , from which we conclude that and giving we deduce that the spatial part of the stationary pdf , to lowest order in , is and the selection procedure is complete , with the result .\label{eq:712}\end{aligned}\ ] ] the solution of the appropriate fokker - planck equation to first order in should , of course , produce the same outcome , but our purpose here is to demonstrate that it can emerge also from considerations of entropy production .we now examine the mean production of , writing and employing eq .( [ eq : s2av ] ) we obtain }{\partial v}\right)^{2},\quad\label{eq:51}\end{aligned}\ ] ] which is consistent with a direct averaging of eq .( [ eq:701 ] ) . inserting eq .( [ eq:707 ] ) we find that ^{2},\qquad\label{eq:51b-1}\end{aligned}\ ] ] to lowest order in and in the stationary state we therefore have similarly , eq .( [ eq : s3av ] ) gives ,\label{eq:51a}\ ] ] which leads to .\quad\label{eq:52 - 3}\end{aligned}\ ] ] this clearly vanishes in the stationary state , and for close to stationarity in the velocity coordinate ( in the sense that , where ) , the mean rate of production is of order .note that the stationary solution to the dynamics ( [ eq:712 ] ) satisfies a local equipartition condition but that this relationship is valid only to first order in , in contrast to the approach of where such a condition is taken to be a requirement to all orders . under the dynamics assumed here ,the system is maintained away from exact local equipartition through the flows of heat between the various local reservoirs .it is instructive to consider next the variational identification of the stationary state under nonisothermal conditions within a framework of _ overdamped _ dynamics , and to contrast the outcome with the analysis in section [ sub : underdamped - dynamics ] for underdamped dynamics .the revised dynamics will affect the form of each component of entropy production , although it remains the case that the principal relaxational component will increase on average until the stationary state is reached .it is well known that eqs .( [ eq:1 ] ) and ( [ eq:1a ] ) reduce for large to the it-rules sde for the position coordinate : together with an associated fokker - planck equation for the positional pdf .we investigate the entropy production implied by these dynamics , focussing attention on where the superscript ` od ' indicates that it is associated with the overdamped dynamics .  for an it-rules sde ^{1/2}dw ] and /dx)dt ] by and write (v) ] .the function is specified by and the normalisation (v)=1 12 & 12#1212_12%12[1][0] __ ( , ) _ _ ( , ) _ _ ( , ) _ _ ( , ) in  _ _ , ( , ) * * ,  ( ) * * ,  ( ) ( ) * * ,  ( ) _ _ ( , )  * * ,  ( ) _ _ ( , ) _ _ ( , ) in  _ _ , ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , ) _ _ ( , ) _ _ ( , ) * * ,  ( ) _ _ , , vol . ( , , ) * * ,  ( ) * * ,  ( ) _ _ ( , ) in http://arxiv:1201.6381v1 [ _ _ ] , ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , ) _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: the selection of an equilibrium state by maximising the entropy of a system , subject to certain constraints , is often powerfully motivated as an exercise in logical inference , a procedure where conclusions are reached on the basis of incomplete information . but such a framework can be more compelling if it is underpinned by dynamical arguments , and we show how this can be provided by stochastic thermodynamics , where an explicit link is made between the production of entropy and the stochastic dynamics of a system coupled to an environment . the separation of entropy production into three components allows us to select a stationary state by maximising the change , averaged over all realisations of the motion , in the principal relaxational or nonadiabatic component , equivalent to requiring that this contribution to the entropy production should become time independent for all realisations . we show that this recovers the usual equilibrium probability density function ( pdf ) for a conservative system in an isothermal environment , as well as the stationary nonequilibrium pdf for a particle confined to a potential under nonisothermal conditions , and a particle subject to a constant nonconservative force under isothermal conditions . the two remaining components of entropy production account for a recently discussed thermodynamic anomaly between over- and underdamped treatments of the dynamics in the nonisothermal stationary state .


the study of automata over data words , i.e. , words over an infinite alphabet , has its origins in the seminal work by kaminski and francez .their finite - memory automata ( more commonly called _register automata _ ) equip finite - state machines with registers in which data values ( from the infinite alphabet ) can be stored and be reused later .register automata preserve some of the good properties of finite automata : they have a decidable emptiness problem and are closed under union and intersection . on the other hand , register automata are neither determinizable nor closed under complementation , and they have an undecidable equivalence / inclusion problem .there are actually several variants of register automata , which all have the same expressive power but differ in the complexity of decision problems . in the sequel ,many more automata models have been introduced ( not necessarily with registers ) , aiming at a good balance between expressivity , decidability , and closure properties .some of those models extend register automata , inheriting their drawbacks such as undecidability of the equivalence problem .we will follow the work on register automata and study a model that supports the notion of _ freshness_. when reading a data value , it may enforce that the data value is _ fresh _ ,i.e. , it has not occurred in the whole history of the run .this feature has been proposed in to model computation with names in the context of programming - language semantics . actually , fresh names are needed to model object creation in object - oriented languages , and they are important ingredients in modeling security protocols which often make use of so - called fresh nonces to achieve their security assertions .fresh names are also crucial in the field of network protocols , and they are one of the key features of the -calculus . like ordinary register automata , fresh - register automata preserve some of the good properties of finite automata .however , they are not closed under complement and also come with an undecidable equivalence problem .  in this paper, we propose _ session automata _ , a robust automata model over data words .like register automata , session automata are a syntactical restriction of fresh - register automata , but in an orthogonal way .register automata drop the feature of checking _ global freshness _ ( referring to the whole history ) while keeping a local variant ( referring to the registers ) .session automata , on the other hand , discard local freshness , while keeping the global one .session automata are well - suited whenever fresh values are important for a finite period , for which they will be stored in one of the registers .they correspond to the model from without stacks .not surprisingly , we will show that session automata and register automata describe incomparable classes of languages of data words , whereas both are strictly weaker than fresh - register automata .contrary to finite - state unification based automata introduced in  , session automata ( like fresh - register automata ) do not have the capability to reset the content of a register .however , they can test global freshness which the model of can not .the _ variable automata _ from do not employ registers , but rather use bound and free variables .however , variable automata are close to our model : they use a finite set of bound variables to track the occurrences of some data values , and a single free variable for all other data values ( that must be different from data values tracked by bound variables ) .contrary to our model , variable automata can not test for global freshness , but we are not able to recognize the language of all data words , contrary to them .  in this paper , we show that session automata ( i ) are closed under intersection , union , and resource - sensitive complementation , ( ii ) have a unique canonical form ( analogous to minimal deterministic finite automata ) , ( iii ) have a decidable equivalence / inclusion problem , and ( iv ) enjoy logical characterizations .altogether , this provides a versatile framework for languages over infinite alphabets .  in a second part of the paper , we present an application of our automata model in the area of learning , where decidability of the equivalence problem is crucial .learning automata deals with the inference of automata based on some partial information , for example samples , which are words that either belong to the accepted language or not .a popular framework is that of active learning defined by angluin in which a learner may consult a teacher for so - called membership and equivalence queries to eventually infer the automaton in question .learning automata has many applications in computer science .notable examples are the use in model checking and testing .see for an overview .while active learning of regular languages is meanwhile well understood and is supported by freely available libraries such as learnlib and libalf , extensions beyond plain regular languages are still an area of active research .recently , automata dealing with potentially infinite data as basis objects have been studied .seminal works in this area are that of and . while the first two use abstraction and refinement techniques to cope with infinite data , the second approach learns a sub - class of register automata .note that session automata are incomparable with the model from .thanks to their closure and decidability properties , a conservative extension of angluin s classical algorithm will do for their automatic inference .the paper is structured as follows . in section [ sec : data ] we introduce session automata . section [ sec : snf - can ] presents the main tool allowing us to establish the results of this paper , namely the use of data words in symbolic normal form and the construction of a canonical session automaton .the section also presents some closure properties of session automata and the decidability of the equivalence problem .section [ sec : logic ] gives logical characterizations of our model . in section [ sec : learning ] , we present an active learning algorithm for session automata .this paper is an extended version of .we let be the set of natural numbers and be the set of non - zero natural numbers . in the following , we fix a non - empty finite alphabet of _ labels _ and an infinite set of _ data values_. in examples , we usually use .a _ data word _ over and is a sequence of pairs . in other words, is an element from . for , we let denote the position where occurs for the first time , i.e. , such that and there is no such that . accordingly , we define to be the last position where occurs .  an example data word over and is given by  .we have and .this section recalls two existing automata models over data words namely register automata , previously introduced in , and fresh - register automata , introduced in as a generalization of register automata .moreover , we introduce the new model of session automata , our main object of interest .  register automata ( initially called finite - memory automata ) equip finite - state machines with registers in which data values can be stored and be read out later .fresh - register automata additionally come with an oracle that can determine if a data value is _ fresh _ ,i.e. , has not occurred in the history of a run . both register and fresh - register automata are closed under union and intersection , and they have a decidable emptiness problem .however , they are not closed under complementation , and their equivalence problem is undecidable , which limits their application in areas such as model checking and automata learning .session automata , on the other hand , are closed under ( resource - sensitive ) complementation , and they have a decidable inclusion / equivalence problem .  given a set , we let , , and . in the automata models that we are going to introduce , will be the set of registers .transitions will be labeled with an element from , which determines a register and the operation that is performed on it .more precisely , writes a globally fresh value into , writes a locally fresh value into , and uses the value that is currently stored in . for ,we let if .similarly ,   a _ fresh - register automaton _ ( over and ) is a tuple where  * is the non - empty finite set of _ states _ , * is the non - empty finite set of _ registers _ , * is the _ initial state _ , * is the set of _ final states _ , and * is a finite set of _ transitions _ : each transition is a tuple of the form where are the source and target state , respectively , , and .we call the _ transition label_.  for a transition , we also write . when taking this transition , the automaton moves from state to state and reads a symbol .if , then is the data value that is currently stored in register .if , then is some _ globally fresh _ data value , which has not been read in the _ whole _ history of the run ; is then written into register . finally ,if , then is some _ locally fresh _ data value , which is _ currently _ not stored in the registers ; it will henceforth be stored in register .let us formally define the semantics of .configuration _ is a triple where is the current state , is a partial mapping encoding the current register assignment , and is the set of data values that have been used so far . by ,we denote the set of registers such that is defined .moreover , .we say that is _ final _ if . as usual , we define a transition relation over configurations and let , where , if there is a transition such that the following conditions hold :  1 . 2 . and , 3 . and for all .a run of on a data word is a sequence for suitable configurations with ( here the partial mapping  represents the mapping with empty domain ) .the run is _ accepting _ if is a final configuration .the _ language _ of is then defined as the set of data words for which there is an accepting run .note that fresh - register automata can not distinguish between data words that are equivalent up to permutation of data values .more precisely , given , we write if and such that , for all , we have iff  . for instance , . in the following , the equivalence class of a data word wrt . is written {\approx}$ ] .we call a _ data language _ if , for all such that , we have if , and only if , . in particular , is a data language for every fresh - register automaton .we obtain natural subclasses of fresh - register automata when we restrict the transition labels in the transitions .  a _ register automaton _ is a fresh - register automaton where every transition label is from .  likeregister automata , session automata are a syntactical restriction of fresh - register automata , but in an orthogonal way . instead of local freshness, they include the feature of global freshness .a _ session automaton _ is a fresh - register automaton where every transition label is from .we first compare the three models of automata introduced above in terms of expressive power .[ ex : req - ack ] consider the set of labels and the set of data values , representing an infinite supply of process identifiers ( pids ) .we model a simple ( sequential ) system where processes can approach a server and make a request , indicated by , and where the server can acknowledge these requests , indicated by .more precisely , means that the process with pid performs a request , which is acknowledged when the system executes .[ fig : multiples3](a ) depicts a register automaton that recognizes the language of data words verifying the following conditions :  * there are at most two open requests at a time ; * a process waits for an acknowledgment before making another request ; * every acknowledgment is preceded by a request ; * requests are acknowledged in the order they are received .  in the figure ,an edge label of the form shall denote that there are two transitions , one labeled with , and one labeled with . whereas a transition labeled with is taken when the current data value does not appear currently in the registers ( but could have appeared before in the data word ) and store it in , transition labeled with simply checks that the current data is stored in register .the automaton models a server that can store two requests at a time and will acknowledge them in the order they are received . for example , it accepts .  when we want to guarantee that , in addition , every process makes at most one request , we need the global freshness operator . [fig : multiples3](b ) hence depicts a session automaton recognizing the language of all the data words of in which every process makes at most one request .notice that the transition from to is now labeled with , so that this transition can only be taken in case the current data value has never been seen before .we obtain from by replacing every occurrence of with . while is no longer contained in ,  still accepted .  as a last example , consider the language of data words in which every process makes at most one request ( without any other condition ) . a fresh - register automaton recognizing it is given in [ fig : fra ] .= 4 ( 0)(0,0) ( 1)(-20,0) ( 2)(20,0) ( 12)(0,13) ( 21)(0,-13) ( 0,1) ( 1,0) ( 1,12) ( 2,21) ( 21,1) ( 12,2) ( 0,2) ( 2,0)  = 4 ( 0)(0,0) ( 1)(-20,0) ( 2)(20,0) ( 12)(0,13) ( 21)(0,-13) ( 0,1) ( 1,0) ( 1,12) ( 2,21) ( 21,1) ( 12,2) ( 0,2) ( 2,0)  [ cols="^,^ " , ]   + :  we apply our learning algorithm on the data language given by the automaton of [ fig : examplecanonical](a ) . in [ fig : learning ] the successive observation tables constructed by the algorithm are given .to save space some letters whose rows contain only s are omitted . in [ fig : hypos ] the successive automata constructed from the closed observation tables are given .for sake of clarity we omit the sink states .we start with the alphabet .we omit letters and .table is obtained after initialization and closing by adding to the top .we use to indicate that all letters will lead to the same row . from first hypothesis automaton is constructed .we suppose that the equivalence query gives back as counterexample the data word whose normal form is . herethe break - point yields the distinguishing word .we add it to .the obtained table is not closed anymore .we close it by adding to the top and get table yielding hypothesis automaton . notice that .this means that the equivalence query must give back a data word whose normal form is using at least registers ( here with normal form ) .as the word uses registers , we extend the alphabet to and obtain table .we close the table and get .from there we obtain the hypothesis automaton . after the equivalence query we get as normal form of the data word counterexample . after adding to and closing the table by moving to the top we get finally the table from which the canonical automaton is obtained and the equivalence query succeeds .in this paper , we developed a theory of session automata , which form a robust class of data languages .in particular , they are closed under union , intersection , and resource - sensitive complementation .moreover , they enjoy logical characterizations in terms of ( a fragment of ) mso logic with a predicate to compare data values for equality . finally , unlike most other automata models for data words , session automata have a decidable inclusion problem .this makes them attractive for verification and learning .in fact , we provided a complete framework for algorithmic learning of session automata , making use of their canonical normal form . an interesting direction to followwould be to try to apply those methods to other models of automata dealing with data values like data automata or variable automata . as a next step ,we plan to employ our setting for various verification tasks .in particular , the next step is to implement our framework , using possibly other learning algorithms than the one of rivest and shapire that we presented in this article , for instance using the learnlib platform or libalf .f. aarts , f. heidarian , h. kuppens , p. olsen , and f. w. vaandrager .automata learning through counterexample guided abstraction refinement . in _ fm _ ,volume 7436 of _ lecture notes in computer science _ , pages 1027 .springer , 2012 .t. berg , o. grinchtein , b. jonsson , m. leucker , h. raffelt , and b. steffen . on the correspondence between conformance testing and regular inference . in _fase _ , volume 3442 of _ lecture notes in computer science _ , pages 175189 .springer , 2005 .b. bollig , a. cyriac , p. gastin , and k. narayan kumar .model checking languages of data words . in l.birkedal , editor , _ proceedings of fossacs12 _ , volume 7213 of _ lecture notes in computer science _, pages 391405 .springer , 2012 .b. bollig , p. habermehl , m. leucker , and b. monmege . a fresh approach to learning register automata . in _ proceedings of the 17th international conference on developments in language theory ( dlt13 ) _ , volume 7907 of _ lecture notes in computer science _ , pages 118130 .springer , 2013 .j. m. cobleigh , d. giannakopoulou , and c. s. pasareanu .learning assumptions for compositional verification . in _ tacas _ , volume 2619 of _ lecture notes in computer science _ ,pages 331346 .springer , 2003 .t. colcombet , c. ley , and g. puppis . on the use of guards for logics with data . in_ proceedings of mfcs11 _ , volume 6907 of _ lecture notes in computer science _ , pages 243255 .springer berlin / heidelberg , 2011 .o. grumberg , o. kupferman , and s. sheinvald .an automata - theoretic approach to reasoning about parameterized systems and specifications . in _volume 8172 of _ lecture notes in computer science _ , pages 397411 .springer , 2013 .a. kurz , t. suzuki , and e. tuosto . on nominal regular languages with binders . in l.birkedal , editor , _ proceedings of fossacs12 _ , volume 7213 of _ lecture notes in computer science _ , pages 255269 .springer , 2012 .

Abstract: we introduce _ session automata _ , an automata model to process data words , i.e. , words over an infinite alphabet . session automata support the notion of _ fresh _ data values , which are well suited for modeling protocols in which sessions using fresh values are of major interest , like in security protocols or ad - hoc networks . session automata have an expressiveness partly extending , partly reducing that of classical register automata . we show that , unlike register automata and their various extensions , session automata are robust : they ( i ) are closed under intersection , union , and ( resource - sensitive ) complementation , ( ii ) admit a symbolic regular representation , ( iii ) have a decidable inclusion problem ( unlike register automata ) , and ( iv ) enjoy logical characterizations . using these results , we establish a learning algorithm to infer session automata through membership and equivalence queries .   = 1