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the problem of the lifetime of a metastable state has been addressed in a variety of areas , including first - order phase transitions , josephson junctions , field theory and chemical kinetics  @xcite . recent experimental and theoretical results show that long - live metastable states are observed in different areas of physics  @xcite . experimental and theoretical investigations have shown that the average escape time from metastable states in fluctuating potentials presents a nonmonotonic behavior as a function of the noise intensity with the presence of a maximum  @xcite . this is the noise enhanced stability ( nes ) phenomenon : the stability of metastable states can be enhanced and the average life time of the metastable state increases nonmonotonically with the noise intensity . this resonance - like behavior contradicts the monotonic behavior of the kramers theory  @xcite . the occurrence of the enhancement of stability of metastable states by the noise has been observed in different physical and biological systems  @xcite . very recently nes effect was observed in an ecological system  @xcite , an oscillator chemical system ( the belousov - zhabotinsky reaction )  @xcite and in magnetic systems  @xcite . interestingly in ref . @xcite the stabilization of a metastable state due to noise is experimentally detected and a decreasing behavior of the maximum lyapunov exponent as a function of the noise intensity is observed . a generalization of the lyapunov exponent for stochastic systems has been recently defined in ref . @xcite to complement the analysis of the transient dynamics of metastable states . this new measure of stability is the mean growth rate coefficient ( mgrc ) @xmath1 and it is evaluated by a similar procedure used for the calculation of the lyapunov exponent in stochastic systems  @xcite . by linearizing the langevin equation of motion ( see next eq .  [ eq ] ) , we consider the evolution of the separation @xmath2 between two neighboring trajectories of the brownian particle starting at @xmath3 and reaching @xmath4    @xmath5    and define @xmath6 as an instantaneous growth rate . we note that , in eq . ( [ delta ] ) , @xmath7 is calculated onto the noisy trajectory @xmath8 $ ]  @xcite . the growth rate coefficient @xmath9 ( for the @xmath10 noise realization ) , is then defined as the long - time average of the instantaneous @xmath11 coefficient over @xmath12  @xcite @xmath13 in the limit @xmath14 , eq . ( [ lyap ] ) coincides formally with the definition of the maximum lyapunov exponent , and therefore , the @xmath9 coefficient has the meaning of a finite - time lyapunov exponent . this quantity is useful to characterize a transient dynamics in nonequilibrium dynamical systems  @xcite . the mean growth rate coefficient @xmath1 is then defined as the ensemble average of the growth rate coefficient @xmath9 @xmath15 over the noise realizations . the mean growth rate coefficient has a nonmonotonic behavior as a function of the noise intensity for brownian particles starting from unstable initial positions  @xcite . this nonmonotonicity with a minimum indicates that @xmath1 can be used as a new suitable measure or signature of the nes effect .    the inclusion of realistic noise sources , with a finite correlation time , impacts both the stationary and the dynamic features of nonlinear systems . for metastable thermal equilibrium systems it has been demonstrated that colored thermal noise can substantially modify the crossing barrier process  @xcite . a rich and enormous literature on escape processes driven by colored noise was produced in the @xmath16 s  @xcite . more recently many papers investigated the role of the correlated noise on different physical systems  @xcite , which indicates a renewed interest in the realistic noise source effects . in this work we present a study of the average decay time of an overdamped brownian particle subject to a cubic potential with a metastable state . we focus on the role of different unstable initial conditions and of colored noise in the average escape time . the effect of the correlation time @xmath0 on the transient dynamics of the escape process is related to the characteristic time scale of the system , that is the relaxation time inside the metastable state @xmath17 . for @xmath18 , the dynamical regime of the brownian particle is close to the white noise dynamics . for @xmath19 , we obtain : ( i ) a big shift of the increase of the average escape times towards higher noise intensities ; ( ii ) an enhancement of the value of the average escape time maximum with a broadening of the nes region in the plane ( @xmath20 ) , which becomes very large for high values of @xmath0 ; ( iii ) the shift of the peculiar initial position @xmath21 ( towards lower values ) , found in our previous studies  @xcite , which separates the set of the initial unstable states producing divergency , for @xmath22 tending to zero , from those which give only a nonmonotonic behavior of the average escape time ; ( iv ) the entire qualitative behaviors ( i - iii ) can be applied to the standard deviation of the escape time ; ( v ) the shift of the minimum values in the curves of the mean growth rate coefficient @xmath1 ; ( vi ) trend to the disappearance of the minimum in the curves of @xmath1 , with a decreasing monotonic behavior for increasing @xmath0 ; ( vii ) trend to the disappearance of the divergent dynamical regime in @xmath23 , with increasing @xmath0 . the paper is organized as follows . in the next section we introduce the model . in the third section we show the results and in the final section we draw the conclusions . the starting point of our study is the langevin equation @xmath24 where @xmath25 is the ornstein - uhlenbeck process @xmath26 and @xmath27 is the wiener process with the usual statistical properties : @xmath28 and @xmath29 . the system of eqs . ( [ eq ] ) and  ( [ eqou ] ) represents a two - dimensional markovian process , which is equivalent to a non - markovian langevin equation driven with additive gaussian correlated noise , with @xmath25 obeying the following statistical properties @xmath30 and @xmath31 , for @xmath32 and @xmath33 . here @xmath34 is the correlation time of the process . the integration of eq . ( [ eqou ] ) yields in the limit @xmath35 the white noise term @xmath36 and the stationary correlation function of the ornstein - uhlenbeck process gives in the limit @xmath35 the correlation function of the white noise : @xmath37 . the potential @xmath38 used in eq . ( [ eq ] ) is @xmath39 , with @xmath40 , @xmath41 . the potential profile has a local stable state at @xmath42 and an unstable state at @xmath43 ( see fig . [ 1-fpot ] ) . the relaxation time for the metastable state at @xmath42 is @xmath44_{x=0 } = 2 a$ ] , which is the characteristic time scale of our system . for our potential profile we have @xmath45 . as a function of noise intensity @xmath22 in the case of correlated noise with @xmath46 , for the four initial positions investigated ( see fig .  [ 1-fpot ] ) . inset : the related standard deviation as a function of the noise intensity @xmath22 . the dotted straight line at @xmath47 separates the simulation data representing the brownian particles escaped within the maximum simulation time @xmath48 for @xmath49 , from those representing the particles partially trapped within the well for a time greater or equal to @xmath48 for @xmath50 . panel b : mean growth rate coefficient @xmath1 as a function of the noise intensity @xmath22 , with the same initial positions of fig . [ 1-fpot].,width=321 ]     reaching the threshold position @xmath51 into the potential well , within the @xmath48 , as a function of noise intensity @xmath22 . this threshold position @xmath52 corresponds to the flex point of the potential , where the instantaneous growth rate @xmath6 is equal to zero . the correlation time of the noise is @xmath53 , with the same initial conditions of fig . [ 1-fpot].,width=321 ]    the calculations of the average escape time as a function of the colored noise intensity have been performed by averaging over * @xmath54 * realizations the numerical solution of the stochastic differential equation  ( [ eq ] ) . the absorbing boundary for the escape process is put on @xmath55 , and the maximum simulation time is @xmath56 . for all the initial unstable states ( see fig . [ 1-fpot ] ) and all the correlation times considered we find an enhancement of the mean first passage time ( mfpt ) @xmath23 with respect to the deterministic time .     reaching the threshold position @xmath51 into the potential well , within the @xmath48 , as a function of noise intensity @xmath22 . this threshold position @xmath52 corresponds to the flex point of the potential , where the instantaneous growth rate @xmath6 is equal to zero . the correlation time of the noise is @xmath57 , with the same initial conditions of fig . [ 1-fpot].,width=321 ]    in fig . [ 2-nes ] the calculations performed with low colored noise ( @xmath58 ) for the mean first passage time @xmath23 and the mean growth rate coefficient @xmath1 are shown . we see as signatures of the nes effect a maximum in the curve of @xmath23 and a minimum in that of @xmath1 . in the inset of fig . [ 2-nes]a the standard deviation of the first passage time as a function of noise intensity is reported . we note that the behaviors of @xmath23 and @xmath1 in this low colored noise regime ( @xmath46 ) coincides with those obtained in the white noise case  @xcite . moreover by comparing the theoretical predictions of @xmath23 ( see eq . ( 3 ) of ref . @xcite ) with direct numerical simulations of the langevin equation , a very good agreement is obtained ( see fig .  3 of ref . @xcite ) . in fig . [ 3-n_i ] the semi - log plots of the fraction of particles @xmath59 reaching the threshold position @xmath60 into the potential well , within the @xmath48 , as a function of noise intensity @xmath22 , with the same initial conditions of fig . [ 1-fpot ] , are shown . this threshold position @xmath52 corresponds to the concavity change of the potential and is considered for this reason as a reference indicator for the effective entrance of the particle into the well . it is possible to observe that for very low noise intensity none particle enters into the well within the @xmath48 considered , and the estimation of the stability measures take their deterministic values . we note that the behavior of the mean growth rate coefficient as a function of the noise intensity is strongly affected by the characteristic potential shape of a metastable state . the curves shown in fig .  [ 3-n_i ] clarify the behavior of @xmath1 in the limit of @xmath61 . in fact the position @xmath51 is the flex point of the potential , where the instantaneous growth rate @xmath6 is equal to zero . we see that for low noise intensities the fraction @xmath59 goes to zero , producing an increasing behavior of the mgrc ( see fig .  [ 2-nes]b ) . the behaviors of the mfpts as a function of the noise intensity @xmath22 with other values of @xmath0 are shown in fig . [ 4-tau_c ] . we clearly observe two dynamical regimes depending on the value of @xmath0 with respect to the relaxation time of the system ( @xmath62 ) : ( a ) weak colored noise ( @xmath63 ) and ( b ) strong colored noise ( @xmath64 ) . by observing fig . [ 4-tau_c]a ( @xmath65 ) we can see that the qualitative behavior of mfpt shown in fig . [ 2-nes]a is recovered . in the weak color noise regime we can still observe the divergent behavior of mfpts for @xmath66 and a non monotonic behavior for @xmath67 , with @xmath68 . by increasing the value of the correlation time ( @xmath69 ) we observe a large displacement of the maximum of mfpt towards higher values of noise intensity and a shift of the peculiar initial position @xmath21 towards lower values . for @xmath70 , @xmath71 , and for @xmath72 , @xmath73 ( see figs .  [ 4-tau_c]b and  [ 4-tau_c]c ) , where @xmath74 is the peculiar initial position of the brownian particle in the presence of colored noise . we note that @xmath68 is a fixed value for white noise case  @xcite , while the position @xmath74 is a variable quantity for colored noise and it is depending on the value of the correlation time of the noise . moreover , we observe a broadening of the nes region , which becomes very large for high values of the correlation time @xmath0 . the nes region is the area where enhanced stability of a metastable state is observed . in other words it is the area under each curve of @xmath23 _ vs _ @xmath22 ( see figs .  [ 2-nes]a and  [ 4-tau_c ] ) , where the values of @xmath23 are greater than the deterministic dynamical time related to the particular initial position investigated ( see also fig .  1 in _ mantegna and spagnolo _ , 1998 , ref .  @xcite ) . the asymmetry of the potential profile with respect to the @xmath75 coordinate makes more effective the correlation of the noise for brownian particles moving from left to right . this means that , at very low noise intensities of the colored noise , the particles inside the potential well will escape more easily with respect to the white noise case . therefore , the trapping effect , which is responsible for the divergent behavior for any initial unstable state within the range @xmath76 will happen in a restricted range of initial positions , that is @xmath77 with @xmath78 . specifically this peculiar position @xmath74 is shifted towards decreasing values of the @xmath75 coordinate for increasing correlation time @xmath0 of the noise source . in fig . [ 2-nes]a and all panels of fig . [ 4-tau_c ] the dotted straight line at @xmath47 separates the simulation data representing the brownian particles escaped within the maximum simulation time @xmath48 for @xmath49 , from those representing the particles partially trapped within the well for a time greater or equal to @xmath48 for @xmath50 . this means that the simulation data obtained for @xmath50 underestimate the real data in the divergent dynamical regime . in fact if we prolong the maximum simulation time @xmath48 we obtain more approximate values for @xmath23 and @xmath79 and the divergent behavior will be visible at lower noise intensities . as a consequence @xmath80 will be shifted towards lower values .    for high values of the noise intensity all the plots show a monotonic decrease behavior as a function of noise intensity collapsing in a unique curve . moreover the slope of this limit curve becomes flatter by increasing the correlation time . this means that the nes effect involves more and more orders of magnitude of the noise intensity . the effect of the colored noise is therefore to delay the escape process or in other words to enhance more and more the stability of the metastable state for increasing values of the noise intensity .    in fig . [ 5-sigma ] the standard deviation @xmath79 of the first passage time distribution for @xmath57 is shown . we see a huge increase of the @xmath79 for low values of noise intensity , demonstrating a strong enlargement of the distribution when the particle feels a noise intensity comparable with the height of potential barrier . similarly to the mfpts , _ color _ induces a shift in the divergent behavior of @xmath79 . the relative measure of the width with respect to the mean value is shown in the inset of fig . [ 5-sigma ] , where the ratio @xmath81 is plotted . this ratio reveals a nonmonotonic behavior with a minimum , demonstrating the existence of a noise intensity for which the width of the first passage time distribution is the minimum related to its mean . in other words this value corresponds to a maximum of precision in the measure of @xmath23 . this optimal noise intensity is shifted toward high noise values by increasing @xmath0 .    the behavior of the mean growth rate coefficient @xmath1 as a function of the noise intensity @xmath22 for different values of the noise correlation time is shown in fig . [ 6-lya_c ] . in the weak color noise regime we observe a nonmonotonic behavior with a minimum for all the initial positions investigated with a shift in the position of the minimum towards higher noise intensities . in the strong color regime the minimum , which represents a trapping phenomenon for a finite time , is visible for the divergent behavior of mfpts for @xmath82 and it is shifted towards higher noise intensities by increasing the correlation time . for initial positions @xmath83 , the minimum tends to disappear , but at the same time the @xmath1 parameter decreases monotonically with increasing noise intensity , showing a trapping phenomenon at higher noise intensities . this trend to the disappearance of the minimum in the curves of @xmath1 , corresponds to the restricted range of the initial positions for which we observe a divergent behavior of @xmath23 , that is to the trend of disappearance of this divergent behavior . we note that the behaviour of @xmath1 as a function of the noise intensity @xmath22 obtained in our analysis is in qualitative agreement with that obtained by the experimental investigation of the stabilization of a metastable state in an oscillatory chemical system ( the belousov - zhabotinsky reaction )  @xcite . specifically the decreasing behavior of the maximum lyapunov exponent of fig .  2 of ref . @xcite is in qualitative good agreement with the behavior of the curves in fig .  [ 6-lya_c]b and fig . [ 6-lya_c]c . this could be ascribed to the correlation time always present in noise sources used in any experimental set - up .    in fig . [ 7-n_i_c ] we report , for all the initial positions investigated and for @xmath57 , the semi - log plot of the fraction of particles @xmath59 entering into the potential well up to the position @xmath84 , within the @xmath48 , as a function of noise intensity @xmath22 . at very low noise intensities and for increasing values of the correlation time @xmath0 , the particles have difficulty to enter into the potential well , within the @xmath48 considered , shifting the entrance statistics towards higher values of the noise intensity . in this work we analyzed the effect of the colored noise , generated by an ornstein - uhlenbeck process , on the enhancement of the mean first passage time in a cubic potential with a metastable state and on the minimum of the mean growth rate coefficient as a function of the noise intensity . we analyze different initial unstable states . we obtain nes effect for all the initial positions investigated and an enhancement of the nes region for increasing values of correlation times . the results obtained for a particle moving in a cubic potential are quite general , because we always obtain nes effect when a particle is initially located just to the right of a local potential maximum and next to a metastable state , in the escape region .    in experiments real noise sources are correlated with a finite correlation time . as a consequence the nes effect can be observed at higher noise intensities with respect to the idealized white noise case . the enhancement and the shift of the nes region , towards higher values of the noise intensity , allows to reveal experimentally the nes effect only by using a suitable correlation time @xmath0 in the noise source . i. biazzo , f. caltagirone , g. parisi , and f. zamponi , phys . lett . * 102 * , 195701 ( 2009 ) ; r. giacchetti and e. sorace , phys . * 101 * , 190401 ( 2008 ) ; l. m. krauss and j. dent , phys . . lett . * 100 * , 171301 ( 2008 ) . n. v. agudov , b. spagnolo , phys . e * 64 * , 035102(r ) ( 2001 ) ; n. v. agudov , a. a. dubkov , b. spagnolo , physica a * 325 * , 144 ( 2003 ) ; b. spagnolo , a. a. dubkov , n. v. agudov , acta physica polonica b , vol . * 35 * ( 4 ) , 1419 ( 2004 ) . l. schimansky - geier and h. herzel , j. stat . phys . * 70 * , 141 ( 1993 ) ; g. paladin , m. serva , and a. vulpiani , phys . lett . * 74 * , 66 ( 1995 ) ; v. loreto , g. paladin , and a. vulpiani , phys . e * 53 * , 2087 ( 1996 ) .
we present a study of the escape time from a metastable state of an overdamped brownian particle , in the presence of colored noise generated by ornstein - uhlenbeck process . 
 we analyze the role of the correlation time on the enhancement of the mean first passage time through a potential barrier and on the behavior of the mean growth rate coefficient as a function of the noise intensity . 
 we observe the noise enhanced stability effect for all the initial unstable states used , and for all values of the correlation time @xmath0 investigated . 
 we can distinguish two dynamical regimes characterized by weak and strong correlated noise respectively , depending on the value of @xmath0 with respect to the relaxation time of the system .