Alp Gorker

Hasan Erdem Malazgirt
email: alp.malazgirt@boun.edu.tr

Arda Yantır

Yurdakul
email: yurdakul@boun.edu.tr

Smail Niar
email: smail.niar@univ-valenciennes.fr

Application Specific Multi-port Memory Customization in FPGAs

FPGA block RAMs (BRAMs) offer speed advantages compared to LUT-based memory designs but a BRAM has only one read and one write port. Designers need to use multiple BRAMs in order to create multi-port memory structures which are more difficult than designing with LUT-based multiport memories. Multi-port memory designs increase overall performance but comes with area cost. In this paper, we present a fully automated methodology that tailors our multi-port memory from a given application. We present our performance improvements and area tradeoffs on state-of-the-art string matching algorithms.

I. INTRODUCTION

In FPGAs, multi-port memories are extensively used in soft processors and custom hardware accelerators for increasing performance. Current mainstream soft processors in FPGAs are RISC processors but for extensive parallelism in FPGAs, soft VLIW processors are also suggested. Although VLIW processors allow more parallelism, they are difficult to optimize because of larger memory requirements than RISC processors. This is due to increasing port numbers, code size and register usage for supporting more parallelism. Performance improvement through exploiting parallelism has costs. Software designers need to parallelize sequential applications. Hardware designers need design exploration in order to utilize hardware resources and design hardware that suits the requirements of given applications.

In this paper, we present a fully automated methodology that tailors the multi-port memory from a given application for a generic VLIW processor that can be customized for a given application. This method leverages instruction level parallelism of memory and ALU operations from a sequential algorithm's execution trace. Given an application, we automatically extract parallelism from execution traces to determine its required input output port numbers that leverages parallelism. Then, we reorder the traces and bundle them and simulate the performance with the FPGA technology file that supplies the technology characteristics such as delay, area and power for an estimation of resource utilization and execution speed of the application.

II. RELATED WORK A. Extracting Instruction Level Parallelism (ILP)

Parallelism extraction and parallelizing instructions have been applied widely in compiler domain [START_REF] Fisher | Trace scheduling: A technique for global microcode compaction[END_REF][2][3] [START_REF] Hsu | Highly concurrent scalar processing[END_REF]. Par-allelizing compilers for VLIW processors provides ILP by encapsulating more than one operations in one instruction at compile time. Software pipelining, trace scheduling, predicated execution, hierarchical reduction and speculative execution have been major compiler optimizations. Trace scheduling [START_REF] Fisher | Trace scheduling: A technique for global microcode compaction[END_REF] requires additional code when operations are reordered. Speculative execution [START_REF] Smith | Boosting beyond static scheduling in a superscalar processor[END_REF] reduces the compensation code and moves instructions past branches. Software pipelining [START_REF] Lam | Software pipelining: An effective scheduling technique for vliw machines[END_REF]aims at compacting loop kernels by minimizing initiation intervals. Predicated execution [START_REF] Hsu | Highly concurrent scalar processing[END_REF] aims to convert branches to basic blocks with hardware defined predicates on certain instructions. Hierarchical reduction [START_REF] Lam | Software pipelining: An effective scheduling technique for vliw machines[END_REF] is the method to simplify rescheduling process by compacting and representing scheduled program components as a single component. Approaches that extract ILP parallelism without using execution traces require dependency analysis methods such as points-to-analysis [START_REF] Wilson | Efficient context-sensitive pointer analysis for C programs[END_REF] which is computationally expensive. Our method analyzes execution traces where all the address dependencies can be checked via real memory addresses. Register dependencies can be checked via register locations. Control Flow Graph (CFG) is extracted and used to determine loop bounds and inter loop dependencies.

B. Multi-port Memories in FPGAs

The most efficient method of designing multi-port memories in FPGAs is using block RAMs (BRAM) that are available in traditional FPGAs [START_REF]Xilinx, IP Processor Block RAM (BRAM) Block[END_REF] [START_REF]Altera, Internal Memory (RAM and ROM) User Guide[END_REF]. However, a BRAM has only two ports, which are not sufficient to implement a highly parallel architecture that usually needs multiple ports. The third method is combining on-chip BRAMs in conventional methods to form a multi-port memory. In multi-port memory design with BRAMs, two common methods are replication and banking. Replication and banking can be exploited in order to increase the number of read and write ports [START_REF] Saghir | A configurable multi-ported register file architecture for soft processor cores[END_REF] [START_REF] Yantir | Efficient implementations of multi-pumped multi-port register files in fpgas[END_REF]. However in this method, memory is partitioned between the banks. Multiple-read can be done from the same bank, but two or more write operations cannot be done on the same bank. In order to handle this problem, a set of software and hardware methods were proposed by Anjam et al [START_REF] Anjam | A multiported register file with register renaming for configurable softcore vliw processors[END_REF] [START_REF] Laforest | Efficient multi-ported memories for fpgas[END_REF]. In this work, we have used the multi-port memory design suggested by [START_REF] Yantir | Efficient implementations of multi-pumped multi-port register files in fpgas[END_REF] because it has shown to provide better multi-port performance explained in the work.

III. APPLICATION ANALYSIS AND PARALLELISM EXTRACTION

Execution traces are generated from the application binary and the input data set. Our trace generation program uses PIN [START_REF] Luk | Pin: building customized program analysis tools with dynamic instrumentation[END_REF] library and application binaries are compiled from C/C++ source files.

For parallelism analysis, we have derived rules for extracting parallelism from a sequential algorithm. These rules reorder and bundle instructions without breaking data dependencies. For preventing structural hazards, we assume that there are sufficient number of internal registers in the system and register renaming is employed to prevent structural hazards. The details of parallelism extraction are explained in our previous work [START_REF] Malazgirt | Mipt: Rapid exploration and evaluation for migrating sequential algorithms to multiprocessing systems with multi-port memories[END_REF].

Instructions that are bundled together are parallel and can be scheduled at the same access step. This reordering extracts the maximum parallelism which is defined as the highest number of instructions in an access step.

Direct implementation of a multi-port memory which can support maximum parallelism will be inefficient in terms of area and power consumption though it provides the fastest execution time. Our recommendation algorithm described in [START_REF] Malazgirt | Mipt: Rapid exploration and evaluation for migrating sequential algorithms to multiprocessing systems with multi-port memories[END_REF] reduces maximum port usage and issue slot size and keeps maximum parallelism. Hence, we gain significantly from resource usage while keeping the performance attained at maximum parallelism.

IV. MULTI-PORT MEMORY ARCHITECTURE

In traditional FPGA architectures, there exist dedicated BRAMs for storage [START_REF]Xilinx, IP Processor Block RAM (BRAM) Block[END_REF] [START_REF]Altera, Internal Memory (RAM and ROM) User Guide[END_REF]. These BRAMs have two ports which are used either as write or read port depending on the BRAM mode. In true dual port mode, each port can change its read/write behavior during run time i.e. 2R or 1R & 1W or 2W. In simple dual port mode, a BRAM has exactly dedicated one read and one write port and configuration cannot be changed during run time.

We have synthesized 64x512 bit memory with minimum delay constraints and reported post-place and route maximum clock period. The minimum delay constraints are reduced iteratively by our automation tool. At each iteration, our tool lowers the minimum delay constraint and synthesizes the design. The process continues until the minimum delay constraint is not met with 0.01 ns difference.

The results have been generated on Xilinx Virtex-5 XC5VLX110T and Zynq-7020. Figure 1a and Figure 1b show that increasing number of write ports decreases memory access speed more than increasing the number of read ports for Virtex-5. Thus, though increasing the number of ports might guarantee the required parallelism, it might degrade performance due to unexpected slowdown of memory accesses. Zynq-7020 has also shown similar behavior therefore we only present Virtex-5 results.

V. EXPERIMENTAL RESULTS

We developed a custom simulator in order to compare the performance of our multi-port memory and available We apply our method on string matching algorithms shown in Table I. String matching has been used extensively and different approaches have been developed. Nevertheless, all of them are highly memory intensive [START_REF] Faro | The exact online string matching problem: A review of the most recent results[END_REF]. Some of the algorithms we experiment have different variations. In FSBNDMQ, we change the q grams and the selected lookahead values. In FAOSO, alignment numbers vary. In TVSBS, we used different window sizes.

Input set contains one million characters of human DNA. In this text, we search for a pattern with a length of ten thousand characters and each character is one byte. Hence a pattern is 10kB long. The text alphabet size is four. Loading data from external memory to FPGA memory is handled by custom logic controller which controls a multi-paged memory architecture. Initially, all the pages are filled with data. The number of active pages and idle pages are determined according to the algorithms' behavior. Idle pages are loaded with new data. This mechanism allows memory references to take constant amount of cycles. In addition, we assume that the change of input data set doesn't affect the algorithm behavior. All our multi-port, true dual port and simple dual port memory configurations have 32 bits word size and depth is 4096. Each multi-port memory with 32*4096 bit configuration allows us to store a pattern, because this configuration can store 16KB of data. Xilinx Zynq-7020 has 560KB of memory [START_REF]Xilinx, Zynq-7000 All Programmable SoC Technical Reference Manual[END_REF], therefore our paged memory architecture can load up to 560KB/16KB = 35 different pages.

Performance of the string matching algorithms of multi issue multi-port (MP) memory are compared with single issue true dual port (TDP) and single issue simple dual port configurations (SDP). The obtained execution times are represented as e M P , e T DP and e SDP respectively in Table II. Operation costs for non-memory operations are taken from [START_REF]Intel, Intel Software Developer Manuals[END_REF]. Issue size of non-memory operations are limited to the maximum number of read/write port number. For example, 4R 1W configuration is assumed to have less than or equal to five issue slots.

All of the algorithms have shown speed ups between 1.55x to 4.25x in MP over TDP and SDP memories shown in Table II. In addition, experiments show that algorithms with higher average parallelism do not necessarily yield better performance. FSBNDMQ algorithm has the least average parallelism and least ratio of memory instructions among all algorithms. FAOSO6 has the largest number of read/write ports and TVSBS-w8 has the highest average parallelism. However, they are both slower than FSBNDMQ. The average parallelism does not seem to correlate with access step values. The average Fig. 2: Area-Delay product of multi-port memory configuration normalized to true dual port parallelism of FJS algorithm is 4.13 whereas FSBNDMQ implementations have 2.85 on average. Nevertheless, FJS has six times more access steps than FSBNDMQ. Similarly, read/write port numbers do not convey a hint about the recommended access steps. GRASPM has lower number of read/write ports than FAOSO6, but it requires more access steps to complete.

The number of read and write ports affect FPGA utilization differently. SDP has the lowest resource usage. TDP employs more BRAMs than SDP. Depending on the number of read ports, MP configurations use more BRAMs than TDP and SDP. All configurations with single write port use only one LUT. MP with multiple write ports use more LUTs and BRAMs than equivalent single port configurations. For example, 3R 2W configuration consumes more BRAMs and LUTs than 4R 1W configuration.

Figure 2 presents the area-delay product of multi-port configurations which are normalized to true dual port configurations. Results show that majority of multi-port configurations are more efficient than dual port. However, algorithms with multiple write ports are inefficient because the usage of BRAM and LUT is very high compared to dual port memory configuration.

VI. CONCLUSION

In this work, we present a method which extracts instruction level parallelism from sequential algorithm and tailors our multi-port memory by recommending multi-port memory size and issue slot size. Our custom simulator perform performance estimations comparing our multi-port memory over true dual and simple dual port memory on FPGAs. Our method has improved FSBDMQ algorithm by 3x with Area-Delay product of 0.7 compared to true dual port BRAM memory.

Fig. 1 :

 1 Fig. 1: Memory access speed change with respect to read and write ports of 64x512 bit memory on Virtex-5

ACKNOWLEDGMENT

  This work is supported by Bogazici University Scientific Research Projects (BAP) Project No: 11A01P6 and by e M P /e T DP e M P /e SDP

TABLE I :

 I String Matching Algorithms and abbreviations

TABLE II :

 II R & W port sizes, average parallelism, FPGA resource utilizations and simulation time of string matching algorithms TUBITAK given to Prof. S. NIAR under Project No: 2221.


S Joubaud

B Percier

A Petrosyan

S Ciliberto

Aging and effective temperatures near a critical point

Keywords: numbers: 05.40.-a, 05.70.Jk, 02.50.-r, 64.60.-i

The orientation fluctuations of the director of a liquid crystal (LC) are measured after a quench near the Fréedericksz transition, which is a second order transition driven by an electric field. We report experimental evidence that, because of the critical slowing down, the LC presents several properties of an aging system after the quench, such as power law scaling in times of correlation and response functions. During this slow relaxation, a well defined effective temperature, much larger than the heat bath temperature, can be measured using the fluctuation dissipation relation.

The characterization of the thermodynamic properties of out of equilibrium and of slow relaxing systems is an important problem of great current interest, which is studied both theoretically and experimentally. In this framework an important and general question, which has been analyzed only theoretically is the relaxation of a system, which is rapidly quenched exactly at the critical point of a second order phase transition [1,2,3,4,5]. Because of the well known divergence of the relaxation time and of the correlation length the system presents a very rich dynamic, which has been named "aging at the critical point". Indeed this relaxation dynamic presents several properties, which are reminiscent of those observed during the aging of spin glasses [1,2,3,4,5,[START_REF] Cugliandolo | Dynamics of glassy systems[END_REF][START_REF] Vincent | Ageing and the Glass Transition[END_REF] : the mean quantity has an algebraic decay and the correlation and response functions exhibit a power law scaling in time. Another important feature analyzed during the aging at a critical point concerns the equilibrium relation between response and correlation, i.e. the fluctuation dissipation theorem (FDT), which is not necessarily satisfied in an out of equilibrium system. This statement is relevant in the context of spin glass aging where a well defined (in the thermodynamic sense) effective temperature T eff can be obtained using the so called fluctuation dissipation relation (FDR). This relation leads to the introduction of the fluctuation dissipation ratio, X, between correlation and response functions [START_REF] Cugliandolo | Dynamics of glassy systems[END_REF][START_REF] Cugliandolo | [END_REF]. Deviations of X from unity (X = 1 is the equilibrium value) may quantify the distance from equilibrium [START_REF] Cugliandolo | [END_REF]. Equivalently an effective temperature T eff = T /X, larger than the heat bath temperature T , can be defined. Those ideas have been transferred to the aging at critical point, where for certain variables the FDR can be interpreted as a well defined effective temperature T eff [1,2].

The study of these analogies between the aging at the critical point and the aging of spin glasses is important because it allows to give new insight on the role of the quenched disorder on the above mentioned features and on the common mechanisms producing them. However these theoretical studies have been performed mainly on spin models and most importantly the reduced control parameter ǫ has been set exactly equal to zero. Thus one is interested to know how general these predictions are and also one may wonder whether those predictions can be observed in an experimental system where the exact condition ǫ = 0 can never be reached.

The purpose of this letter is to analyze, within this theoretical framework, an experiment on the relaxation dynamic close to the critical point of a liquid crystal instability, which is, at a first approximation, described by a Ginzburg-Landau equation and has the main features of a second order phase transition. The main result of our investigation is that after a quench close to the critical point the effective temperature T eff , measured from FDR, has a well defined value larger than the thermal bath temperature.

This kind of experimental test is important because the properties of X and T eff have attracted much interest, since they suggest that a generalized statistical mechanics can be defined for a broad class of non equilibrium phenomena. Several theoretical models or numerical simulations show such a behavior [START_REF] Cugliandolo | Dynamics of glassy systems[END_REF]9]. However in experiments the results are less clear and the relevance for real materials of the T eff as defined by FDR is still an open question, which merits investigation [10,11,12,13]. For example, there are important differences, which are not understood, between supercooled fluids [10], polymers [12], gels [11] and spin glasses [13]. The point is that on this subject it is very difficult to find simple theoretical models, which can be directly compared with experiments. Thus the experimental study of the relaxation dynamics close to the critical point is very useful in this sense.

The system of our interest is the Fréedericksz transition of a liquid crystal (LC), subjected to an electric field E [START_REF] De Gennes | The physics of liquid crystals[END_REF][START_REF] Oswald | Nematic and cholesteric liquid crystals[END_REF]. In this system, we measure the variable ζ, which is the spatially averaged alignment of the LC molecules, whose local direction of alignment is defined by the unit pseudo vector n. Let us first recall the general properties of the Fréedericksz transition. The system under consideration is a LC confined between two parallel glass plates at a distance L = 9 µm. The inner surfaces of the confining plates have transparent Indium-Tin-Oxyde (ITO) electrodes, used to apply the electric field. Furthermore the plate surfaces, are coated by a thin layer of polymer mechanically rubbed in one direction. This surface treatment causes the alignment of the LC molecules in a unique direction parallel to the surface (planar alignment),i.e. all the molecules have the same director parallel to the x-axis [16]. The cell is next filled by a LC having a positive dielectric anisotropy (p-pentylcyanobiphenyl, 5CB, produced by Merck). The LC is subjected to an electric field perpendicular to the plates, by applying a voltage V between the ITO electrodes, i.e. E = V /L. To avoid the electrical polarization of the LC, we apply an AC voltage at a frequency [START_REF] De Gennes | The physics of liquid crystals[END_REF][START_REF] Oswald | Nematic and cholesteric liquid crystals[END_REF]. More details on the experimental set-up can be found in ref. [START_REF] Joubaud | [END_REF]18]. When V 0 exceeds a threshold value V c the planar state becomes unstable and the LC molecules, except those anchored to the glass surfaces, try to align parallel to the field, i.e. the director, away from the confining plates, acquires a component parallel to the applied electric field (z-axis). This is the Fréedericksz transition whose properties are those of a second order phase transition [START_REF] De Gennes | The physics of liquid crystals[END_REF][START_REF] Oswald | Nematic and cholesteric liquid crystals[END_REF]. For V 0 close to V c the motion of the director is characterized by its angular displacement θ in xz-plane (θ is the angle between the x axis and n), whose space-time dependence has the following form : θ = θ 0 (x, y, t) sin πz L [START_REF] De Gennes | The physics of liquid crystals[END_REF][START_REF] Oswald | Nematic and cholesteric liquid crystals[END_REF]19]. If θ 0 remains small then its dynamics, neglecting the (x, y) dependence of θ 0 , is described by a Ginzburg-Landau equation :

f V = 1 kHz (V = √ 2V 0 cos(2πf V t))
τ 0 dθ 0 dt = ǫ θ 0 - 1 2 (κ + ǫ + 1)θ 3 0 + η (1)
where ǫ =

V 2 0 V 2 c
-1 is the reduced control parameter. The characteristic time τ 0 and constant k depend on the LC material properties. η is a thermal noise delta-correlated in time [19].

We define the variable ζ as the spatially averaged alignment of the LC molecules and more precisely :

ζ = 2 L 1 A A dxdy L 0 (1 -n 2 x )dz (2)
where A = πD 2 0 /4 is the area, in the (x, y) plane, of the measuring region of diameter D 0 , which is about 2 mm in our case. If θ 0 remains small, ζ takes a simple form in terms of θ 0 , i.e. n x = cos(θ) and ζ = 1/A A dxdyθ 2 0 . It has been shown that ζ is characterized by a mean value ζ ∝ ǫ, by a divergent relaxation time τ = τ o /ǫ and by fluctuations, which have a Lorentzian spectrum [18,21]. The measure of the variable ζ relies upon the anisotropic properties of the LC, i.e. the cell is a birefringent plate whose local optical axis is parallel to n. This optical anisotropy can be precisely measured using a polarization interferometer [20], which has a signal to noise ratio larger than 1000 (see ref. [START_REF] Joubaud | [END_REF]18] for details).

In this letter, we consider the dynamics of ζ as a function of time after a quench from ǫ = ǫ 1 > 0.3 to ǫ = ǫ 0 << 0.1. The fact that the control parameter is the electric field allows one to have extremely fast quenches, typically 1 ms. As the typical relaxation time τ of ζ is about 10s, at ǫ 0 ≃ 0, this means that we can follow the out of equilibrium dynamics for about four orders of magnitude in time, which is really comparable to what is done in real glass experiments after a temperature quench, where the dynamics is usually followed for about 3 or 4 decades in time. The advantage here is that this relaxation dynamics, which lasts only a few seconds, allows us to repeat the experiment several times and to perform an ensemble average (indicated by < • >) of the measured quantities. We consider first the specific case of a quench from ǫ 1 = 0.3 to ǫ 0 = 0.01. The typical mean value < ζ(t) > of ζ after this quench is plotted in fig. 1 as a function of time t (t = 0 is the time when the quench has been performed). This mean dynamics of ζ is obtained by repeating the same quench 7000 times. The behavior of < ζ > remains constant for a certain time and then slowly relaxes (see fig. 1a). Above a characteristic time, which is about 0.22 s in our case, the relaxation becomes exponential.

To understand this behavior, we decompose the dynamics of θ 0 in a mean dynamics after the quench and its fluctuations, i.e. θ 0 = ψ 0 (t) + δθ 0 , where ψ 0 (t) = θ 0 (t) . The dynamics of Ψ 0 obtained from the analytical solution of eq.1 is shown in fig. 1 as a continuous line, which perfectly agrees with the measured dynamics. The experimental data and the solution of eq.1 have two well distinguish limits : for t ≫ τ ≡ τ 0 /(2ǫ), the relaxation is exponential (τ ≃ 0.22s at ǫ 0 = 0.01) ; for t ≪ τ ≡ τ 0 /(2ǫ 0 ), the dynamics of ψ 0 is almost algebraic

: ψ 0 (t) = (κ+1) τ0 (t + τ m ) -1/2
with τ m = τ0 (κ+1)ψ0(0) 2 . This behavior, plotted in fig. 1 as a dashed line, is identical to the relaxation at the critical point [4]. Thus the system should present aging phenomena in the range t < τ , which is the interval where the dashed line follows FIG. 2: a) Correlations functions C θ (t, tw) as a function of t -tw for different fixed t = 0.06 s (•), 0.2 s ( ), 0.26 s (⋄), 0.28 s (×) and 0.31 s (+) after the same quench of fig. 1. Inset: The correlation functions have a simple master curve obtained by plotting C θ (t, tw)/C θ (t, tw) versus (t-tw)/(t+τm). Notice that (t + τm) is the reduced time used in fig. 1 the experimental data in fig. 1.

In order to measure T eff as defined by the FDR, we need to measure both the correlation function C θ (t, t w ) ≡ δθ 0 (t) δθ 0 (t w ) of the thermal fluctuations δθ 0 of θ 0 and the response function χ(t, t w ) of θ 0 , at time t to a perturbation, given at time t w , to its conjugated variable Γ, with t w < t. More precisely, from ref. [START_REF] Cugliandolo | Dynamics of glassy systems[END_REF][START_REF] Cugliandolo | [END_REF] the effective temperature is given by:

χ(t, t w ) = X(t, t w ) k B T [C θ (t, t) -C θ (t, t w )] (3)
where k B is the Boltzmann constant,T the heat bath temperature, i.e. T eff = T /X(t, t w ).

To compute χ(t, t w ) and C θ (t, t w ) in our experiment one has to consider that the measured variable is ζ. The relationship between the fluctuations of ζ and θ 0 has been discussed in details in ref. [START_REF] Joubaud | [END_REF], thus we recall here only the useful results. As the area of the measuring beam is much larger than the correlation length, the global variable measured by the interferometer is

ζ = 1 A θ 2 0 dxdy ≃ ψ 2 0 + 2ψ 0 δθ 0 . Thus the mean value of ζ is ζ(t) = ψ 2 0 (t)
and the fluctuations of δζ of ζ can be related to the fluctuations of θ 0 : δζ(t) = 2ψ 0 (t)δθ 0 (t).

The autocorrelation function of θ 0 is obtained using the values of ψ 0 (t) and ψ 0 (t w ), i.e. C θ (t, t w ) = ζ(t)ζ(t w ) /(4ψ 0 (t)ψ 0 (t w )). The C θ (t, t w ), measured at various fixed t, are plotted as function of t -t w in fig. 2. We see that a simple scaling of the correlation function can be obtained by plotting them as a function of (t -t w )/(t + τ m ). Thus in agreement with theoretical predictions, our system presents, during its slow relaxation after the quench close to the critical point, algebraic decay of mean quantities and the scaling of correlation functions, which are very similar to the main features of spin glass aging (see for example [START_REF] Cugliandolo | Dynamics of glassy systems[END_REF][START_REF] Vincent | Ageing and the Glass Transition[END_REF]).

The response function is obtained by applying a small change of the voltage V 0 , which modifies the control parameter, i.e. ǫ = ǫ 0 + δǫ. In ref. [START_REF] Joubaud | [END_REF] we have shown that the external torque Γ ext , i.e. the conjugated variable of θ 0 , associated to δǫ is equal to :

Γ ext = 2B δǫ ψ 0 (t) 1 - ψ 0 (t) 2 2 (4)
We separate θ 0 (t) into the average part ψ 0 (t), solution of eq. 1, with η = 0, and a deviation ∆(t) due to δǫ : θ 0 (t) = ψ 0 (t) + ∆(t). We define the linear response function R(t, t w ) =< ∆ > /Γ ext of θ 0 to Γ ext using a Dirac delta function for δǫ at an instant t w , i.e. δǫ(t) ∝ δ(t -t w ). In the experiment the Dirac delta function is approximated by a triangular function of amplitude δǫ 0 and duration τ r ≃ 2ms << τ , specifically:

δǫ = δǫ 0 (1 -2|t -t w |/τ r for |t -t w | < τ r /2 and δǫ = 0 for |t -t w | > τ r /2, with τ r /τ ≃ 10 -4 . The measured quantity is the response R ζ,δǫ =< ∆ζ > /δǫ, of ζ(t) to δǫ(t w ).
As ∆ζ(t) = 2ψ 0 (t)∆ it follows that the linear response of θ 0 to Γ ext is:

R(t, t w ) = R ζ,δǫ (t, t w ) 4B ψ 0 (t w ) ψ 0 (t) 1 -ψ0(tw) 2 2 (5)
where B = Aπ 2 K 1 /(4L), and K 1 a LC elastic constant. Thus inserting in eq.5 the measured values of R ζ,δǫ (t, t w ), of ψ 0 (t) and of ψ 0 (t w ) we can measure R(t, t w ) and by numerical integration we finally obtain χ(t, t w ) = tw t R(t, t ′ )dt ′ . The accuracy of this procedure has been checked at equilibrium in ref. [START_REF] Joubaud | [END_REF], where we have shown that the measured C θ0 and χ(t, t w ) verify FDT.

The integrated response is related to the correlation by the FDR relations (see eq. 3). Due to the non-equilibrium process, it is not equivalent whether the parameter is the waiting time t w or the observation time t (see ref. [9] for a detailed discussion on this point). Applying the correct procedure [2,[START_REF] Cugliandolo | [END_REF], we keep the time t constant and we vary t w , i.e. 0 < t w < t, and we repeat the procedure at various t. Following ref. [START_REF] Cugliandolo | [END_REF], we study the FDR during the relaxation process by plotting the integrated response χ as a function of C θ . These FDR plots can be seen in fig. 3 for four characteristic values of t. When the system is in equilibrium the FDR plot is a straight line with slope -1/(k B T ) (continuous line in fig. 3). The curves in fig. 3 for each t are composed by two straight lines and are remarkably similar to those predicted in theoretical models [START_REF] Cugliandolo | [END_REF] and seen in experiments [13] of spin glasses aging. When t w is close to t, i.e. large values of C(t, t w ), the FDT is satisfied, i.e. the experimental points are on the continuous line in fig. 3. For C(t, t w ) smaller than a value C * , which depends on t, the FDT is not satisfied. However χ(t, t w ) remains linear in C(t, t w ) but the slope, which does not depend on t, is much smaller than its value at equilibrium. This means that the slow modes of the systems have a very well defined T ef f , independent of t, higher than the temperature of the bath. Precisely we find, for C(t, t w ) < C * , X = T /T eff = 0.31, which is remarkably close to the asymptotic value X = 0.305 analytically [5] and numerically [2] estimated for the small wave vector modes of an Ising model quenched at the critical point using a different quenching procedure. This analogy is important because our measurement, being an integral over the measuring volume, is more sensitive to the fluctuations of the long wavelength modes. In fig. 3 we see that C * is a decreasing function of t. Let us define t * as the characteristic time associated with C * . In the inset of fig. 3, we have plotted the value of t * as a function of t. We find that t * ≃ 0 for t ≤ τ = 0.22s. This indicates that, in agreement with ref. [2,4,5], the violation occurs for all times at ǫ 0 = 0. In our case it is cut by the exponential relaxation, which starts at t ≃ τ . For t > τ , t * = t -τ is linear in t, which also agrees with the theoretical picture of the FDR [START_REF] Cugliandolo | [END_REF]. Indeed t * separates the equilibrium part and the aging one and the ratio t * /t = 1 -τ /t, for t > τ , defines how large is the equilibrium interval with respect to the total time. Notice that at ǫ 0 = 0, τ = ∞ and the equilibrium interval does not exist [2]. Thus the values of ǫ 0 and ǫ 1 affect only the amplitude of the region where the out of equilibrium is observed but not the value of X [22]. We also want to stress that this very clear scenario, with a well defined T eff independent of t, is obtained only if the right procedure with fixed t is used [2,9]. The results are completely different if t w is kept fixed. In such a case T eff is a decreasing function of t w [22]. The non-commutability of the two procedures, i.e. either with t or with t w fixed, shows the non ergodicity of the phenomenon and opens a wide discussion on how to analyze the data in more complex situations, such as those where T ef f is measured in real materials and the procedure with t w fixed is often used [10,11,12,13].

In conclusion we have shown that a LC quenched close to the critical point of the Fréedericksz transition presents aging features, such as a power law scaling of correlation functions and the appearance of a well defined T eff > T . Most importantly we show the existence of a well defined T eff if the the right procedure with fixed t is used. What is very interesting here is that although we are not exactly at ǫ 0 = 0 we observe a large interval of time,0 < t < τ where the predicted aging at critical point can be observed. The results plotted in fig. 3 agree with those of ref. [2,5] but are different from those predicted for mean-field [4]. This opens the discussion for further theoretical and experimental developments. It also shows that the study of the quench at critical point is an interesting and not completely understood problem by itself [2,4].

We acknowledge useful discussion with P. Holdsworth, L. Berthier, M. Henkel and M. Pleimling . This work has been partially supported by ANR-05-BLAN-0105-01.

FIG. 1

 1 FIG. 1: a) Mean relaxation of the system, log ζ(t) , as a function of time after a quench (•) from ǫ1 = 0.3 to ǫ0 = 0.01. The continuous line represents the theoretical response based on eq. 1 with ζ ≃ ψ 2 0 . b) Mean relaxation of the system as a function of the reduced time (t + τm). in both figures the dashed lines represents the theoretical relaxation at ǫ = 0.

FIG. 3 :

 3 FIG.3: a) Parametric plot of integrated response versus correlation for t = 0.4 s (•), 0.5 s ( ), 0.6 s (⋄) and 0.7 s (+). Continuous line are linear fits. Dark line represents the FDT at equilibrium. Inset : the characteristic time t * is plotted as a function of t.